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## G = C112order 121 = 112

### Elementary abelian group of type [11,11]

Aliases: C112, SmallGroup(121,2)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C112
 Chief series C1 — C11 — C112
 Lower central C1 — C112
 Upper central C1 — C112
 Jennings C1 — C112

Generators and relations for C112
G = < a,b | a11=b11=1, ab=ba >

Smallest permutation representation of C112
Regular action on 121 points
Generators in S121
```(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 50 51 52 53 54 55)(56 57 58 59 60 61 62 63 64 65 66)(67 68 69 70 71 72 73 74 75 76 77)(78 79 80 81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96 97 98 99)(100 101 102 103 104 105 106 107 108 109 110)(111 112 113 114 115 116 117 118 119 120 121)
(1 121 32 87 98 43 19 76 65 54 109)(2 111 33 88 99 44 20 77 66 55 110)(3 112 23 78 89 34 21 67 56 45 100)(4 113 24 79 90 35 22 68 57 46 101)(5 114 25 80 91 36 12 69 58 47 102)(6 115 26 81 92 37 13 70 59 48 103)(7 116 27 82 93 38 14 71 60 49 104)(8 117 28 83 94 39 15 72 61 50 105)(9 118 29 84 95 40 16 73 62 51 106)(10 119 30 85 96 41 17 74 63 52 107)(11 120 31 86 97 42 18 75 64 53 108)```

`G:=sub<Sym(121)| (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)(45,46,47,48,49,50,51,52,53,54,55)(56,57,58,59,60,61,62,63,64,65,66)(67,68,69,70,71,72,73,74,75,76,77)(78,79,80,81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120,121), (1,121,32,87,98,43,19,76,65,54,109)(2,111,33,88,99,44,20,77,66,55,110)(3,112,23,78,89,34,21,67,56,45,100)(4,113,24,79,90,35,22,68,57,46,101)(5,114,25,80,91,36,12,69,58,47,102)(6,115,26,81,92,37,13,70,59,48,103)(7,116,27,82,93,38,14,71,60,49,104)(8,117,28,83,94,39,15,72,61,50,105)(9,118,29,84,95,40,16,73,62,51,106)(10,119,30,85,96,41,17,74,63,52,107)(11,120,31,86,97,42,18,75,64,53,108)>;`

`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)(45,46,47,48,49,50,51,52,53,54,55)(56,57,58,59,60,61,62,63,64,65,66)(67,68,69,70,71,72,73,74,75,76,77)(78,79,80,81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120,121), (1,121,32,87,98,43,19,76,65,54,109)(2,111,33,88,99,44,20,77,66,55,110)(3,112,23,78,89,34,21,67,56,45,100)(4,113,24,79,90,35,22,68,57,46,101)(5,114,25,80,91,36,12,69,58,47,102)(6,115,26,81,92,37,13,70,59,48,103)(7,116,27,82,93,38,14,71,60,49,104)(8,117,28,83,94,39,15,72,61,50,105)(9,118,29,84,95,40,16,73,62,51,106)(10,119,30,85,96,41,17,74,63,52,107)(11,120,31,86,97,42,18,75,64,53,108) );`

`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),(45,46,47,48,49,50,51,52,53,54,55),(56,57,58,59,60,61,62,63,64,65,66),(67,68,69,70,71,72,73,74,75,76,77),(78,79,80,81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96,97,98,99),(100,101,102,103,104,105,106,107,108,109,110),(111,112,113,114,115,116,117,118,119,120,121)], [(1,121,32,87,98,43,19,76,65,54,109),(2,111,33,88,99,44,20,77,66,55,110),(3,112,23,78,89,34,21,67,56,45,100),(4,113,24,79,90,35,22,68,57,46,101),(5,114,25,80,91,36,12,69,58,47,102),(6,115,26,81,92,37,13,70,59,48,103),(7,116,27,82,93,38,14,71,60,49,104),(8,117,28,83,94,39,15,72,61,50,105),(9,118,29,84,95,40,16,73,62,51,106),(10,119,30,85,96,41,17,74,63,52,107),(11,120,31,86,97,42,18,75,64,53,108)]])`

C112 is a maximal subgroup of   C11⋊D11  C112⋊C3

121 conjugacy classes

 class 1 11A ··· 11DP order 1 11 ··· 11 size 1 1 ··· 1

121 irreducible representations

 dim 1 1 type + image C1 C11 kernel C112 C11 # reps 1 120

Matrix representation of C112 in GL2(𝔽23) generated by

 13 0 0 1
,
 8 0 0 18
`G:=sub<GL(2,GF(23))| [13,0,0,1],[8,0,0,18] >;`

C112 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(121,2);`
`// by ID`

`G=gap.SmallGroup(121,2);`
`# by ID`

`G:=PCGroup([2,-11,11]:ExponentLimit:=1);`
`// Polycyclic`

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

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