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G = C112⋊C4order 484 = 22·112

The semidirect product of C112 and C4 acting faithfully

Aliases: C112⋊C4, C11⋊D11.C2, SmallGroup(484,8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C112 — C112⋊C4
 Chief series C1 — C112 — C11⋊D11 — C112⋊C4
 Lower central C112 — C112⋊C4
 Upper central C1

Generators and relations for C112⋊C4
G = < a,b,c | a11=b11=c4=1, ab=ba, cac-1=a3b6, cbc-1=a2b8 >

121C2
2C11
2C11
2C11
2C11
2C11
2C11
121C4
22D11
22D11
22D11
22D11
22D11
22D11

Permutation representations of C112⋊C4
On 22 points - transitive group 22T8
Generators in S22
```(1 2 3 4 5 6 7 8 9 10 11)(12 13 14 15 16 17 18 19 20 21 22)
(1 5 9 2 6 10 3 7 11 4 8)(12 21 19 17 15 13 22 20 18 16 14)
(1 14)(2 12 11 16)(3 21 10 18)(4 19 9 20)(5 17 8 22)(6 15 7 13)```

`G:=sub<Sym(22)| (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22), (1,5,9,2,6,10,3,7,11,4,8)(12,21,19,17,15,13,22,20,18,16,14), (1,14)(2,12,11,16)(3,21,10,18)(4,19,9,20)(5,17,8,22)(6,15,7,13)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22), (1,5,9,2,6,10,3,7,11,4,8)(12,21,19,17,15,13,22,20,18,16,14), (1,14)(2,12,11,16)(3,21,10,18)(4,19,9,20)(5,17,8,22)(6,15,7,13) );`

`G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11),(12,13,14,15,16,17,18,19,20,21,22)], [(1,5,9,2,6,10,3,7,11,4,8),(12,21,19,17,15,13,22,20,18,16,14)], [(1,14),(2,12,11,16),(3,21,10,18),(4,19,9,20),(5,17,8,22),(6,15,7,13)])`

`G:=TransitiveGroup(22,8);`

34 conjugacy classes

 class 1 2 4A 4B 11A ··· 11AD order 1 2 4 4 11 ··· 11 size 1 121 121 121 4 ··· 4

34 irreducible representations

 dim 1 1 1 4 type + + + image C1 C2 C4 C112⋊C4 kernel C112⋊C4 C11⋊D11 C112 C1 # reps 1 1 2 30

Matrix representation of C112⋊C4 in GL4(𝔽89) generated by

 18 56 0 0 33 78 0 0 0 0 53 11 0 0 78 33
,
 0 1 0 0 88 71 0 0 0 0 78 33 0 0 56 18
,
 0 0 1 0 0 0 0 1 1 0 0 0 71 88 0 0
`G:=sub<GL(4,GF(89))| [18,33,0,0,56,78,0,0,0,0,53,78,0,0,11,33],[0,88,0,0,1,71,0,0,0,0,78,56,0,0,33,18],[0,0,1,71,0,0,0,88,1,0,0,0,0,1,0,0] >;`

C112⋊C4 in GAP, Magma, Sage, TeX

`C_{11}^2\rtimes C_4`
`% in TeX`

`G:=Group("C11^2:C4");`
`// GroupNames label`

`G:=SmallGroup(484,8);`
`// by ID`

`G=gap.SmallGroup(484,8);`
`# by ID`

`G:=PCGroup([4,-2,-2,-11,11,8,3026,246,2691,3527]);`
`// Polycyclic`

`G:=Group<a,b,c|a^11=b^11=c^4=1,a*b=b*a,c*a*c^-1=a^3*b^6,c*b*c^-1=a^2*b^8>;`
`// generators/relations`

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