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## G = C122order 144 = 24·32

### Abelian group of type [12,12]

Aliases: C122, SmallGroup(144,101)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C122
 Chief series C1 — C2 — C22 — C2×C6 — C62 — C6×C12 — C122
 Lower central C1 — C122
 Upper central C1 — C122

Generators and relations for C122
G = < a,b | a12=b12=1, ab=ba >

Subgroups: 90, all normal (6 characteristic)
C1, C2 [×3], C3 [×4], C4 [×6], C22, C6 [×12], C2×C4 [×3], C32, C12 [×24], C2×C6 [×4], C42, C3×C6 [×3], C2×C12 [×12], C3×C12 [×6], C62, C4×C12 [×4], C6×C12 [×3], C122
Quotients: C1, C2 [×3], C3 [×4], C4 [×6], C22, C6 [×12], C2×C4 [×3], C32, C12 [×24], C2×C6 [×4], C42, C3×C6 [×3], C2×C12 [×12], C3×C12 [×6], C62, C4×C12 [×4], C6×C12 [×3], C122

Smallest permutation representation of C122
Regular action on 144 points
Generators in S144
```(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)(121 122 123 124 125 126 127 128 129 130 131 132)(133 134 135 136 137 138 139 140 141 142 143 144)
(1 141 91 40 121 112 25 56 97 19 81 66)(2 142 92 41 122 113 26 57 98 20 82 67)(3 143 93 42 123 114 27 58 99 21 83 68)(4 144 94 43 124 115 28 59 100 22 84 69)(5 133 95 44 125 116 29 60 101 23 73 70)(6 134 96 45 126 117 30 49 102 24 74 71)(7 135 85 46 127 118 31 50 103 13 75 72)(8 136 86 47 128 119 32 51 104 14 76 61)(9 137 87 48 129 120 33 52 105 15 77 62)(10 138 88 37 130 109 34 53 106 16 78 63)(11 139 89 38 131 110 35 54 107 17 79 64)(12 140 90 39 132 111 36 55 108 18 80 65)```

`G:=sub<Sym(144)| (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)(121,122,123,124,125,126,127,128,129,130,131,132)(133,134,135,136,137,138,139,140,141,142,143,144), (1,141,91,40,121,112,25,56,97,19,81,66)(2,142,92,41,122,113,26,57,98,20,82,67)(3,143,93,42,123,114,27,58,99,21,83,68)(4,144,94,43,124,115,28,59,100,22,84,69)(5,133,95,44,125,116,29,60,101,23,73,70)(6,134,96,45,126,117,30,49,102,24,74,71)(7,135,85,46,127,118,31,50,103,13,75,72)(8,136,86,47,128,119,32,51,104,14,76,61)(9,137,87,48,129,120,33,52,105,15,77,62)(10,138,88,37,130,109,34,53,106,16,78,63)(11,139,89,38,131,110,35,54,107,17,79,64)(12,140,90,39,132,111,36,55,108,18,80,65)>;`

`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)(121,122,123,124,125,126,127,128,129,130,131,132)(133,134,135,136,137,138,139,140,141,142,143,144), (1,141,91,40,121,112,25,56,97,19,81,66)(2,142,92,41,122,113,26,57,98,20,82,67)(3,143,93,42,123,114,27,58,99,21,83,68)(4,144,94,43,124,115,28,59,100,22,84,69)(5,133,95,44,125,116,29,60,101,23,73,70)(6,134,96,45,126,117,30,49,102,24,74,71)(7,135,85,46,127,118,31,50,103,13,75,72)(8,136,86,47,128,119,32,51,104,14,76,61)(9,137,87,48,129,120,33,52,105,15,77,62)(10,138,88,37,130,109,34,53,106,16,78,63)(11,139,89,38,131,110,35,54,107,17,79,64)(12,140,90,39,132,111,36,55,108,18,80,65) );`

`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),(121,122,123,124,125,126,127,128,129,130,131,132),(133,134,135,136,137,138,139,140,141,142,143,144)], [(1,141,91,40,121,112,25,56,97,19,81,66),(2,142,92,41,122,113,26,57,98,20,82,67),(3,143,93,42,123,114,27,58,99,21,83,68),(4,144,94,43,124,115,28,59,100,22,84,69),(5,133,95,44,125,116,29,60,101,23,73,70),(6,134,96,45,126,117,30,49,102,24,74,71),(7,135,85,46,127,118,31,50,103,13,75,72),(8,136,86,47,128,119,32,51,104,14,76,61),(9,137,87,48,129,120,33,52,105,15,77,62),(10,138,88,37,130,109,34,53,106,16,78,63),(11,139,89,38,131,110,35,54,107,17,79,64),(12,140,90,39,132,111,36,55,108,18,80,65)])`

C122 is a maximal subgroup of
C122.C2  C12.57D12  C122⋊C2  C126Dic6  C12.25Dic6  C12216C2  C124D12  C1226C2  C1222C2  C122.C3  C42⋊He3

144 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 4A ··· 4L 6A ··· 6X 12A ··· 12CR order 1 2 2 2 3 ··· 3 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

144 irreducible representations

 dim 1 1 1 1 1 1 type + + image C1 C2 C3 C4 C6 C12 kernel C122 C6×C12 C4×C12 C3×C12 C2×C12 C12 # reps 1 3 8 12 24 96

Matrix representation of C122 in GL2(𝔽13) generated by

 4 0 0 6
,
 7 0 0 11
`G:=sub<GL(2,GF(13))| [4,0,0,6],[7,0,0,11] >;`

C122 in GAP, Magma, Sage, TeX

`C_{12}^2`
`% in TeX`

`G:=Group("C12^2");`
`// GroupNames label`

`G:=SmallGroup(144,101);`
`// by ID`

`G=gap.SmallGroup(144,101);`
`# by ID`

`G:=PCGroup([6,-2,-2,-3,-3,-2,-2,216,439]);`
`// Polycyclic`

`G:=Group<a,b|a^12=b^12=1,a*b=b*a>;`
`// generators/relations`

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