## Groups of order less than 16

Section 7.10 outlines the classification of all groups of order less than 16.

Order 2 :   Z2
Order 3 :   Z3
Order 4 :   Z4;   Z2 × Z2
Order 5 :   Z5
Order 6 :   Z6;   S3
Order 7 :   Z7
Order 8 :   Z8;   Z4 × Z2;   Z2 × Z2 × Z2;   D4;   Q
Order 9 :   Z9;   Z3 × Z3
Order 10 :   Z10;   D5
Order 11 :   Z11
Order 12 :   Z12;   Z6 × Z2;   A4;   D6;   Z3 Z4
Order 13 :   Z13
Order 14 :   Z14;   D7
Order 15 :   Z15

We end the list at order 15 because at order 16 the number of nonisomorphic groups is quite a bit larger: in addition to 5 abelian groups (which are easily classified), there are 9 nonabelian groups. At the end of this page is a list giving the number of nonisomorphic groups of each order from 1 to 200. Note that groups of order 10 or less are presented via generators and relations at the end of Chapter 3.

The abelian groups of order 11 through 15 can be classified using the Fundamental Theorem of Finite Abelian Groups. They are cyclic, with one exception: the direct product Z6×Z2 of a cyclic group of order 6 and a cyclic group of order 2.

The nonabelian groups in this range are the dihedral groups D6 and D7, of order 12 and 14 (respectively), together with the alternating group A4, and the semidirect product Z3Z4 of a cyclic group of order 4 acting on a cyclic group of order 3. (In several textbooks, the last group is referred to simply as T.)

At this point it is useful to extend the definition of the general linear group over a field to include the general linear group GLn(R) over any commutative ring R. This is the group of all invertible nxn matrices with entries from the ring R. In particular, we will use the groups GL2(Zn) and GL3(Zn).

The nonabelian groups of order 12 are given below, using HTML table and superscript commands.

#### D6, the dihedral group of order 12

Defined as the group of rigid motions of a regular hexagon, the dihedral group of order 12 is generated by the following permutations of the vertices of the hexagon.
```
a = (1,2,3,4,5,6)        b = (2,6)(3,5)

```

Since D6 is a semidirect product of a cyclic group of order 2 acting on a cyclic group of order 6, it can also be described as the subgroup of GL2(Z6) generated by these matrices.

```     _       _           _       _
|         |         |         |
|  1   0  |         |  1   0  |
a = |         |     b = |         |
|  1   1  |         |  0  -1  |
|_       _|         |_       _|

```

The multiplication table for D6 (given below) uses generators a,b
with relations a6 = 1, b2 = 1, ba = a-1b:

1 a a2 a3 a4 a5 b ab a2b a3b a4b a5b
1 1 a a2 a3 a4 a5 b ab a2b a3b a4b a5b
a a a2 a3 a4 a5 1 ab a2b a3b a4b a5b b
a2 a2 a3 a4 a5 1 a a2b a3b a4b a5b b ab
a3 a3 a4 a5 1 a a2 a3b a4b a5b b ab a2b
a4 a4 a5 1 a a2 a3 a4b a5b b ab a2b a3b
a5 a5 1 a a2 a3 a4 a5b b ab a2b a3b a4b
b b a5b a4b a3b a2b ab 1 a5 a4 a3 a2 a
ab ab b a5b a4b a3b a2b a 1 a5 a4 a3 a2
a2b a2b ab b a5b a4b a3b a2 a 1 a5 a4 a3
a3b a3b a2b ab b a5b a4b a3 a2 a 1 a5 a4
a4b a4b a3b a2b ab b a5b a4 a3 a2 a 1 a5
a5b a5b a4b a3b a2b ab b a5 a4 a3 a2 a 1

Elements:
order 6: a, a5
order 3: a2, a4
order 2: a3, b, ab, a2b, a3b, a4b, a5b

Conjugacy classes:
{1}
{a3}
{a,a5}
{a2,a4}
{b,a2b,a4b}
{ab,a3b,a5b}

Subgroups:
order 12: {1,a,a2,a3,a4,a5, b,ab,a2b,a3b,a4b,a5b}
order 6: {1,a,a2,a3,a4,a5}, {1,a2,a4,b,a2b,a4b}, {1,a2,a4,ab,a3b,a5b}
order 4: {1,a3,b,a3b}, {1,a3,ab,a4b}, {1,a3,a2b,a5b}
order 3: {1,a2,a4}
order 2: {1,a3}, {1,b}, {1,ab}, {1,a2b}, {1,a3b}, {1,a4b}, {1,a5b}
order 1: {1}

Normal subgroups:
order 12: {1,a,a2,a3,a4,a5, b,ab,a2b,a3b,a4b,a5b}
order 6: {1,a,a2,a3,a4,a5}, {1,a2,a4,b,a2b,a4b}, {1,a2,a4,ab,a3b,a5b}
order 3: {1,a2,a4}
order 2: {1,a3}
order 1: {1}

#### Z3Z4, the semidirect product of a cyclic group of order 4 acting on a cyclic group of order 3

Since Z3Z4 is a semidirect product of a cyclic group of order 4 acting on a cyclic group of order 3, it can be described as the subgroup of GL2(Z15) generated by by the following matrices.

```     _       _           _       _
|         |         |         |
|  1   0  |         |  1   0  |
x = |         |     y = |         |
|  5   1  |         |  0   2  |
|_       _|         |_       _|

```
In this presentation the element x has order 3. In most texts the group is described as having two generators, one of order 6 and one of order 4. This can be realized by using the following matrices, where a=x2y2 and x=a2.
```     _       _           _       _
|         |         |         |
|  1   0  |         |  1   0  |
a = |         |     b = |         |
| 10   4  |         |  0   2  |
|_       _|         |_       _|

```

Described via generators a,b
with relations a6 = 1, b2 = a3, ba = a-1b:

1 a a2 a3 a4 a5 b ab a2b a3b a4b a5b
1 1 a a2 a3 a4 a5 b ab a2b a3b a4b a5b
a a a2 a3 a4 a5 1 ab a2b a3b a4b a5b b
a2 a2 a3 a4 a5 1 a a2b a3b a4b a5b b ab
a3 a3 a4 a5 1 a a2 a3b a4b a5b b ab a2b
a4 a4 a5 1 a a2 a3 a4b a5b b ab a2b a3b
a5 a5 1 a a2 a3 a4 a5b b ab a2b a3b a4b
b b a5b a4b a3b a2b ab a3 a2 a 1 a5 a4
ab ab b a5b a4b a3b a2b a4 a3 a2 a 1 a5
a2b a2b ab b a5b a4b a3b a5 a4 a3 a2 a 1
a3b a3b a2b ab b a5b a4b 1 a5 a4 a3 a2 a
a4b a4b a3b a2b ab b a5b a 1 a5 a4 a3 a2
a5b a5b a4b a3b a2b ab b a2 a 1 a5 a4 a3

Note that aib2=ai+3 and aib3=ai+3b.

Elements:
order 6: a, a5
order 4: b, ab, a2b, a3b, a4b, a5b
order 3: a2, a4
order 2: a3

Conjugacy classes:
{1}
{a3}
{a,a5}
{a2,a4}
{b,a2b,a4b}
{ab,a3b,a5b}

Subgroups:
order 12: {1,a,a2,a3,a4,a5, b,ab,a2b,a3b,a4b,a5b}
order 6: {1,a,a2,a3,a4,a5}
order 4: {1,b,a3,a3b}, {1,ab,a3,a4b}, {1,a2b,a3,a5b}
order 3: {1,a2,a4}
order 2: {1,a3}
order 1: {1}

Normal subgroups:
order 12: {1,a,a2,a3,a4,a5, b,ab,a2b,a3b,a4b,a5b}
order 6: {1,a,a2,a3,a4,a5}
order 3: {1,a2,a4}
order 2: {1,a3}
order 1: {1}

As noted above, the group can also be presented as follows,
via generators x,y
with relations x3=1, y4=1, yx=x-1y:

1 x x2 y xy x2y y2 xy2 x2y2 y3 xy3 x2y3
1 1 x x2 y xy x2y y2 xy2 x2y2 y3 xy3 x2y3
x x x2 1 xy x2y y xy2 x2y2 y2 xy3 x2y3 y3
x2 x2 1 x x2y y xy x2y2 y2 xy2 x2y3 y3 xy3
y y x2y xy y2 x2y2 xy2 y3 x2y3 xy3 1 x2 x
xy xy y x2y xy2 y2 x2y2 xy3 y3 x2y3 x 1 x2
x2y x2y xy y x2y2 xy2 y2 x2y3 xy3 y3 x2 x 1
y2 y2 xy2 x2y2 y3 xy3 x2y3 1 x x2 y xy x2y
xy2 xy2 x2y2 y2 xy3 x2y3 y3 x x2 1 xy x2y y
x2y2 x2y2 y2 xy2 x2y3 y3 xy3 x2 1 x x2y y xy
y3 y3 x2y3 xy3 1 x2 x y x2y xy y2 x2y2 xy2
xy3 xy3 y3 x2y3 x 1 x2 xy y x2y xy2 y2 x2y2
x2y3 x2y3 xy3 y3 x2 x 1 x2y xy y x2y2 xy2 y2

With this presentation, the elements and subgroups are as follows.

Elements:
order 6: x2y2, xy2
order 4: y, xy, x2y, y3, xy3, x2y3
order 3: x, x2
order 2: y2

Conjugacy classes:
{1}
{y2}
{x,x2}
{xy2,x2y2}
{y,xy,x2y}
{y3,xy3,x2y3}

Subgroups:
order 12: {1,x,x2,y,xy,x2y,y2,xy2,x2y2,y3,xy3,x2y3}
order 6: {1,x2y2,x,y2,x2,xy2}
order 4: {1,y,y2,y3}, {1,xy,y2,xy3}, {1,x2y,y2,x2y3}
order 3: {1,x,x2}
order 2: {1,y2}
order 1: {1}

Normal subgroups:
order 12: {1,x,x2,y,xy,x2y,y2,xy2,x2y2,y3,xy3,x2y3}
order 6: {1,x2y2,x,y2,x2,xy2}
order 3: {1,x,x2}
order 2: {1,y2}
order 1: {1}

#### A4, the alternating group on 4 elements

Described via generators a,b,c
with relations a2 = 1, b2 = 1, c3 = 1, ba=ab, ca = abc, cb=ac:

1 a b ab c ac bc abc c2 ac2 bc2 abc2
1 1 a b ab c ac bc abc c2 ac2 bc2 abc2
a a 1 ab b ac c abc bc ac2 c2 abc2 bc2
b b ab 1 a bc abc c ac bc2 abc2 c2 ac2
ab ab b a 1 abc bc ac c abc2 bc2 ac2 c2
c c abc ac bc c2 abc2 ac2 bc2 1 ab a b
ac ac bc c abc ac2 bc2 c2 abc2 a b 1 ab
bc bc ac abc c bc2 ac2 abc2 c2 b a ab 1
abc abc c bc ac abc2 c2 bc2 ac2 ab 1 b a
c2 c2 bc2 abc2 ac2 1 b ab a c bc abc ac
ac2 ac2 abc2 bc2 c2 a ab b 1 ac abc bc c
bc2 bc2 c2 ac2 abc2 b 1 a ab bc c ac abc
abc2 abc2 ac2 c2 bc2 ab a 1 b abc ac c bc

Elements:
order 3: c, ac, bc, abc, c2, ac2, bc2, abc2
order 2: a, b, ab

Conjugacy classes:
{1}
{a,b,ab}
{c,ac,bc,abc}
{c2,ac2,bc2,abc2}

Subgroups:
order 12: {1,a,b,ab,c,ac,bc,abc,c2,ac2, bc2,abc2}
order 4: {1,a,b,ab}
order 3: {1,c,c2}, {1,ac,bc2}, {1,bc,abc2}, {1,abc,ac2}
order 2: {1,a}, {1,b}, {1,ab}
order 1: {1}

Normal subgroups:
order 12: {1,a,b,ab,c,ac,bc,abc,c2,ac2, bc2,abc2}
order 4: {1,a,b,ab}
order 1: {1}

The group A4 can be described as the semidirect product of a cyclic group of order 3 acting on the Klein 4 group. To realize this as a matrix group, we can use a subgroup of GL3(Z2). It can be verified that the following matrices satisfy the identities of A4, and so the subgroup they generate is isomorphic to A4.

```     _           _          _           _          _           _
|             |        |             |        |             |
|  1   0   0  |        |  1   0   0  |        |  1   0   0  |
|             |        |             |        |             |
a = |  1   1   0  |    b = |  0   1   0  |    c = |  0   1   1  |
|             |        |             |        |             |
|  0   0   1  |        |  1   0   1  |        |  0   1   0  |
|_           _|        |_           _|        |_           _|

```

The next table presents A4 as the set of even permutations on four elements. It can be realized as the set of rigid motions of a tetrahedron.

1 (12)(34) (13)(24) (14)(23) (123) (243) (142) (134) (132) (143) (234) (124)
1 1 (12)(34) (13)(24) (14)(23) (123) (243) (142) (134) (132) (143) (234) (124)
(12)(34) (12)(34) 1 (14)(23) (13)(24) (243) (123) (134) (142) (143) (132) (124) (234)
(13)(24) (13)(24) (14)(23) 1 (12)(34) (142) (134) (123) (243) (234) (124) (132) (143)
(14)(23) (14)(23) (13)(24) (12)(34) 1 (134) (142) (243) (123) (124) (234) (143) (132)
(123) (123) (134) (243) (142) (132) (124) (143) (234) 1 (14)(23) (12)(34) (13)(24)
(243) (243) (142) (123) (134) (143) (234) (132) (124) (12)(34) (13)(24) 1 (14)(23)
(142) (142) (243) (134) (123) (234) (143) (124) (132) (13)(24) (12)(34) (14)(23) 1
(134) (134) (123) (142) (243) (124) (132) (234) (143) (14)(23) 1 (13)(24) (12)(34)
(132) (132) (234) (124) (143) 1 (13)(24) (14)(23) (12)(34) (123) (142) (134) (243)
(143) (143) (124) (234) (132) (12)(34) (14)(23) (13)(24) 1 (243) (134) (142) (123)
(234) (234) (132) (143) (124) (13)(24) 1 (12)(34) (14)(23) (142) (123) (243) (134)
(124) (124) (143) (132) (234) (14)(23) (12)(34) 1 (13)(24) (134) (243) (123) (142)

Elements:
order 3: (123), (243), (142), (134), (132), (143), (234), (124)
order 2: (12)(34), (13)(24), (14)(23)

Conjugacy classes:
{1}
{(12)(34), (13)(24), (14)(23)}
{(123), (243), (142), (134)}
{(132), (143), (234), (124)}

Subgroups:
order 12: {1,(12)(34),(13)(24),(14)(23),(123),(243),(142),(134),(132),(143), (234),(124)}
order 4: {1,(12)(34),(13)(24),(14)(23)}
order 3: {1,(123),(132)}, {1,(243),(234)}, {1,(142),(124)}, {1,(134),(143)}
order 2: {1,(12)(34)}, {1,(13)(24)}, {1,(14)(23)}
order 1: {1}

Normal subgroups:
order 12: {1,(12)(34),(13)(24),(14)(23),(123),(243),(142),(134),(132),(143), (234),(124)}
order 4: {1,(12)(34),(13)(24),(14)(23)}
order 1: {1}

## The number of nonisomorphic groups of a given order

```
order	 #	 order	 #	 order 	 #	 order	 #

1	 1	51	 1	101	 1	151	 1
2	 1	52	 5	102	 4	152	12
3	 1	53	 1	103	 1	153	 2
4	 2	54	15	104	14	154	 4
5	 1	55	 2	105	 2	155	 2
6	 2	56	13	106	 2	156	18
7	 1	57	 2	107	 1	157	 1
8	 5	58	 2	108	45	158	 2
9	 2	59	 1	109	 1	159      1
10	 2	60	13	110	 6	160    238
11	 1	61	 1	111	 2	161      1
12	 5	62	 2	112	43	162     55
13	 1	63	 4	113	 1	163      1
14	 2	64     267	114	 6	164      5
15	 1	65	 1	115	 1	165      2
16 	14	66	 4	116	 5	166      2
17	 1	67	 1	117	 4	167      1
18	 5	68	 5	118	 2	168     57
19	 1	69	 1	119	 1	169      2
20	 5	70	 4	120	47	170      4
21	 2	71	 1	121	 2	171      5
22	 2	72	50	122	 2	172      4
23	 1	73	 1	123	 1	173      1
24	15	74	 2	124	 4	174      4
25	 2	75	 3	125	 5	175      2
26	 2	76	 4	126	16	176     42
27	 5	77	 1	127	 1	177      1
28	 4	78	 6	128   2328	178      2
29	 1	79	 1	129	 2	179      1
30	 4	80	52	130	 4	180     37
31	 1	81	15	131	 1	181      1
32	51	82	 2	132	10	182      4
33	 1	83	 1	133	 1	183      2
34	 2	84	15	134	 2	184     12
35	 1	85	 1	135	 5	185      1
36	14	86	 2	136	15	186      6
37	 1	87	 1	137	 1	187      1
38	 2	88	12	138	 4	188      4
39	 2	89	 1	139	 1	189     12
40	14	90	10	140	11	190      4
41	 1	91	 1	141	 1	191      1
42	 6	92	 4	142	 2	192      ?
43	 1	93	 2	143	 1	193      1
44	 4	94	 2	144    197	194      2
45	 2	95	 1	145	 1	195      2
46	 2	96     230	146	 2	196     12
47	 1	97	 1	147	 6	197      1
48	52	98	 5	148	 5	198     10
49	 2	99	 2	149	 1	199      1
50	 5     100	16	150	13	200     52

```