Group 8.1 - Z₈

Name	Z₈
Order	8
Exponent	8
Cyclic	yes
Abelian	yes
Elementary	no
Abelian Invariants	[ 8 ]
p-Group	yes (2)
Rank	1
Nilpotent	yes
Supersoluble	yes
Soluble	yes
Simple	no
Perfect	no

Permutation Representation

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

Degree	8
Transitive	yes
Primitive	no
Regular	yes

Cayley Table:

	1	2	3	4	5	6	7	8
1	1	2	3	4	5	6	7	8
2	2	3	4	5	6	7	8	1
3	3	4	5	6	7	8	1	2
4	4	5	6	7	8	1	2	3
5	5	6	7	8	1	2	3	4
6	6	7	8	1	2	3	4	5
7	7	8	1	2	3	4	5	6
8	8	1	2	3	4	5	6	7

Elements:

1 = ()
2 = (1,2,3,4,5,6,7,8)
3 = (1,3,5,7)(2,4,6,8)
4 = (1,4,7,2,5,8,3,6)
5 = (1,5)(2,6)(3,7)(4,8)
6 = (1,6,3,8,5,2,7,4)
7 = (1,7,5,3)(2,8,6,4)
8 = (1,8,7,6,5,4,3,2)

Centre:	8.1 = < (1,2,3,4,5,6,7,8) >
Commutator Subgroup:	1
Frattini Subgroup:	4.1 = < (1,3,5,7)(2,4,6,8) >

Composition Series

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

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

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

Chief Series

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

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

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

Maximal Subgroups

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