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## G = D112order 484 = 22·112

### Direct product of D11 and D11

Aliases: D112, C111D22, C112⋊C22, C11⋊D11⋊C2, (C11×D11)⋊C2, SmallGroup(484,9)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C112 — D112
 Chief series C1 — C11 — C112 — C11×D11 — D112
 Lower central C112 — D112
 Upper central C1

Generators and relations for D112
G = < a,b,c,d | a11=b2=c11=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

11C2
11C2
121C2
2C11
2C11
2C11
2C11
2C11
121C22
11D11
11C22
11C22
11D11
22D11
22D11
22D11
22D11
22D11
11D22
11D22

Permutation representations of D112
On 22 points - transitive group 22T9
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 12)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 16)(9 15)(10 14)(11 13)
(1 3 5 7 9 11 2 4 6 8 10)(12 21 19 17 15 13 22 20 18 16 14)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 12)(10 13)(11 14)```

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

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

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

`G:=TransitiveGroup(22,9);`

49 conjugacy classes

 class 1 2A 2B 2C 11A ··· 11J 11K ··· 11AI 22A ··· 22J order 1 2 2 2 11 ··· 11 11 ··· 11 22 ··· 22 size 1 11 11 121 2 ··· 2 4 ··· 4 22 ··· 22

49 irreducible representations

 dim 1 1 1 2 2 4 type + + + + + + image C1 C2 C2 D11 D22 D112 kernel D112 C11×D11 C11⋊D11 D11 C11 C1 # reps 1 2 1 10 10 25

Matrix representation of D112 in GL4(𝔽23) generated by

 1 0 0 0 0 1 0 0 0 0 14 20 0 0 1 13
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 8 22
,
 14 10 0 0 2 13 0 0 0 0 1 0 0 0 0 1
,
 16 3 0 0 7 7 0 0 0 0 22 0 0 0 0 22
`G:=sub<GL(4,GF(23))| [1,0,0,0,0,1,0,0,0,0,14,1,0,0,20,13],[1,0,0,0,0,1,0,0,0,0,1,8,0,0,0,22],[14,2,0,0,10,13,0,0,0,0,1,0,0,0,0,1],[16,7,0,0,3,7,0,0,0,0,22,0,0,0,0,22] >;`

D112 in GAP, Magma, Sage, TeX

`D_{11}^2`
`% in TeX`

`G:=Group("D11^2");`
`// GroupNames label`

`G:=SmallGroup(484,9);`
`// by ID`

`G=gap.SmallGroup(484,9);`
`# by ID`

`G:=PCGroup([4,-2,-2,-11,-11,246,7043]);`
`// Polycyclic`

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

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