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## G = C62order 36 = 22·32

### Abelian group of type [6,6]

Aliases: C62, SmallGroup(36,14)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62
 Chief series C1 — C3 — C32 — C3×C6 — C62
 Lower central C1 — C62
 Upper central C1 — C62

Generators and relations for C62
G = < a,b | a6=b6=1, ab=ba >

Smallest permutation representation of C62
Regular action on 36 points
Generators in S36
```(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)
(1 15 30 33 8 23)(2 16 25 34 9 24)(3 17 26 35 10 19)(4 18 27 36 11 20)(5 13 28 31 12 21)(6 14 29 32 7 22)```

`G:=sub<Sym(36)| (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), (1,15,30,33,8,23)(2,16,25,34,9,24)(3,17,26,35,10,19)(4,18,27,36,11,20)(5,13,28,31,12,21)(6,14,29,32,7,22)>;`

`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), (1,15,30,33,8,23)(2,16,25,34,9,24)(3,17,26,35,10,19)(4,18,27,36,11,20)(5,13,28,31,12,21)(6,14,29,32,7,22) );`

`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)], [(1,15,30,33,8,23),(2,16,25,34,9,24),(3,17,26,35,10,19),(4,18,27,36,11,20),(5,13,28,31,12,21),(6,14,29,32,7,22)])`

C62 is a maximal subgroup of   C327D4  C32.A4  C32⋊A4

36 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 6A ··· 6X order 1 2 2 2 3 ··· 3 6 ··· 6 size 1 1 1 1 1 ··· 1 1 ··· 1

36 irreducible representations

 dim 1 1 1 1 type + + image C1 C2 C3 C6 kernel C62 C3×C6 C2×C6 C6 # reps 1 3 8 24

Matrix representation of C62 in GL2(𝔽7) generated by

 5 0 0 1
,
 6 0 0 3
`G:=sub<GL(2,GF(7))| [5,0,0,1],[6,0,0,3] >;`

C62 in GAP, Magma, Sage, TeX

`C_6^2`
`% in TeX`

`G:=Group("C6^2");`
`// GroupNames label`

`G:=SmallGroup(36,14);`
`// by ID`

`G=gap.SmallGroup(36,14);`
`# by ID`

`G:=PCGroup([4,-2,-2,-3,-3]);`
`// Polycyclic`

`G:=Group<a,b|a^6=b^6=1,a*b=b*a>;`
`// generators/relations`

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