Copied to
clipboard

## G = C72order 49 = 72

### Elementary abelian group of type [7,7]

Aliases: C72, SmallGroup(49,2)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C72
 Chief series C1 — C7 — C72
 Lower central C1 — C72
 Upper central C1 — C72
 Jennings C1 — C72

Generators and relations for C72
G = < a,b | a7=b7=1, ab=ba >

Smallest permutation representation of C72
Regular action on 49 points
Generators in S49
```(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 16 17 18 19 20 21)(22 23 24 25 26 27 28)(29 30 31 32 33 34 35)(36 37 38 39 40 41 42)(43 44 45 46 47 48 49)
(1 49 41 34 27 20 13)(2 43 42 35 28 21 14)(3 44 36 29 22 15 8)(4 45 37 30 23 16 9)(5 46 38 31 24 17 10)(6 47 39 32 25 18 11)(7 48 40 33 26 19 12)```

`G:=sub<Sym(49)| (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28)(29,30,31,32,33,34,35)(36,37,38,39,40,41,42)(43,44,45,46,47,48,49), (1,49,41,34,27,20,13)(2,43,42,35,28,21,14)(3,44,36,29,22,15,8)(4,45,37,30,23,16,9)(5,46,38,31,24,17,10)(6,47,39,32,25,18,11)(7,48,40,33,26,19,12)>;`

`G:=Group( (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28)(29,30,31,32,33,34,35)(36,37,38,39,40,41,42)(43,44,45,46,47,48,49), (1,49,41,34,27,20,13)(2,43,42,35,28,21,14)(3,44,36,29,22,15,8)(4,45,37,30,23,16,9)(5,46,38,31,24,17,10)(6,47,39,32,25,18,11)(7,48,40,33,26,19,12) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7),(8,9,10,11,12,13,14),(15,16,17,18,19,20,21),(22,23,24,25,26,27,28),(29,30,31,32,33,34,35),(36,37,38,39,40,41,42),(43,44,45,46,47,48,49)], [(1,49,41,34,27,20,13),(2,43,42,35,28,21,14),(3,44,36,29,22,15,8),(4,45,37,30,23,16,9),(5,46,38,31,24,17,10),(6,47,39,32,25,18,11),(7,48,40,33,26,19,12)]])`

C72 is a maximal subgroup of   C7⋊D7  C72⋊C3  C723C3  He7  7- 1+2
C72 is a maximal quotient of   He7  7- 1+2

49 conjugacy classes

 class 1 7A ··· 7AV order 1 7 ··· 7 size 1 1 ··· 1

49 irreducible representations

 dim 1 1 type + image C1 C7 kernel C72 C7 # reps 1 48

Matrix representation of C72 in GL2(𝔽29) generated by

 25 0 0 1
,
 7 0 0 23
`G:=sub<GL(2,GF(29))| [25,0,0,1],[7,0,0,23] >;`

C72 in GAP, Magma, Sage, TeX

`C_7^2`
`% in TeX`

`G:=Group("C7^2");`
`// GroupNames label`

`G:=SmallGroup(49,2);`
`// by ID`

`G=gap.SmallGroup(49,2);`
`# by ID`

`G:=PCGroup([2,-7,7]:ExponentLimit:=1);`
`// Polycyclic`

`G:=Group<a,b|a^7=b^7=1,a*b=b*a>;`
`// generators/relations`

Export

׿
×
𝔽