Group 18.1 - Z₁₈

Name	Z₁₈
Order	18
Exponent	18
Cyclic	yes
Abelian	yes
Elementary	no
Abelian Invariants	[ 2, 9 ]
p-Group	no
Nilpotent	yes
Supersoluble	yes
Soluble	yes
Simple	no
Perfect	no

Permutation Representation

< (1,2), (3,4,5,6,7,8,9,10,11) >

Degree	11
Transitive	no
Primitive	no
Regular	no

Cayley Table:

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
1	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
2	2	3	4	5	6	7	8	9	1	11	12	13	14	15	16	17	18	10
3	3	4	5	6	7	8	9	1	2	12	13	14	15	16	17	18	10	11
4	4	5	6	7	8	9	1	2	3	13	14	15	16	17	18	10	11	12
5	5	6	7	8	9	1	2	3	4	14	15	16	17	18	10	11	12	13
6	6	7	8	9	1	2	3	4	5	15	16	17	18	10	11	12	13	14
7	7	8	9	1	2	3	4	5	6	16	17	18	10	11	12	13	14	15
8	8	9	1	2	3	4	5	6	7	17	18	10	11	12	13	14	15	16
9	9	1	2	3	4	5	6	7	8	18	10	11	12	13	14	15	16	17
10	10	11	12	13	14	15	16	17	18	1	2	3	4	5	6	7	8	9
11	11	12	13	14	15	16	17	18	10	2	3	4	5	6	7	8	9	1
12	12	13	14	15	16	17	18	10	11	3	4	5	6	7	8	9	1	2
13	13	14	15	16	17	18	10	11	12	4	5	6	7	8	9	1	2	3
14	14	15	16	17	18	10	11	12	13	5	6	7	8	9	1	2	3	4
15	15	16	17	18	10	11	12	13	14	6	7	8	9	1	2	3	4	5
16	16	17	18	10	11	12	13	14	15	7	8	9	1	2	3	4	5	6
17	17	18	10	11	12	13	14	15	16	8	9	1	2	3	4	5	6	7
18	18	10	11	12	13	14	15	16	17	9	1	2	3	4	5	6	7	8

Elements:

1 = ()
2 = (3,4,5,6,7,8,9,10,11)
3 = (3,5,7,9,11,4,6,8,10)
4 = (3,6,9)(4,7,10)(5,8,11)
5 = (3,7,11,6,10,5,9,4,8)
6 = (3,8,4,9,5,10,6,11,7)
7 = (3,9,6)(4,10,7)(5,11,8)
8 = (3,10,8,6,4,11,9,7,5)
9 = (3,11,10,9,8,7,6,5,4)
10 = (1,2)
11 = (1,2)(3,4,5,6,7,8,9,10,11)
12 = (1,2)(3,5,7,9,11,4,6,8,10)
13 = (1,2)(3,6,9)(4,7,10)(5,8,11)
14 = (1,2)(3,7,11,6,10,5,9,4,8)
15 = (1,2)(3,8,4,9,5,10,6,11,7)
16 = (1,2)(3,9,6)(4,10,7)(5,11,8)
17 = (1,2)(3,10,8,6,4,11,9,7,5)
18 = (1,2)(3,11,10,9,8,7,6,5,4)

Centre:	18.1 = < (3,4,5,6,7,8,9,10,11), (1,2) >
Commutator Subgroup:	1
Frattini Subgroup:	3.1 = < (3,6,9)(4,7,10)(5,8,11) >

Composition Series

< (3,4,5,6,7,8,9,10,11), (3,6,9)(4,7,10)(5,8,11), (1,2) >

< (3,6,9)(4,7,10)(5,8,11), (1,2) >

Chief Series

< (1,2), (3,4,5,6,7,8,9,10,11) >

< (3,6,9)(4,7,10)(5,8,11), (1,2) >

Maximal Subgroups

< (3,4,5,6,7,8,9,10,11), (3,6,9)(4,7,10)(5,8,11) >

< (1,2), (3,6,9)(4,7,10)(5,8,11) >

Primary Components

2	< (1,2) >
3	< (3,4,5,6,7,8,9,10,11), (3,6,9)(4,7,10)(5,8,11) >