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## G = C12.2D8order 192 = 26·3

### 2nd non-split extension by C12 of D8 acting via D8/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C12.2D8
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C4×C12 — C12⋊2Q8 — C12.2D8
 Lower central C3 — C2×C6 — C2×C12 — C12.2D8
 Upper central C1 — C22 — C42 — C4⋊C8

Generators and relations for C12.2D8
G = < a,b,c | a12=b8=1, c2=a3, bab-1=a7, cac-1=a5, cbc-1=a9b-1 >

Subgroups: 184 in 64 conjugacy classes, 33 normal (31 characteristic)
C1, C2 [×3], C3, C4 [×4], C4 [×3], C22, C6 [×3], C8 [×2], C2×C4 [×3], C2×C4 [×2], Q8 [×2], Dic3 [×2], C12 [×4], C12, C2×C6, C42, C4⋊C4 [×3], C2×C8 [×2], C2×Q8, C3⋊C8, C24, Dic6 [×2], C2×Dic3 [×2], C2×C12 [×3], C4⋊C8, C4⋊C8, C4⋊Q8, C2×C3⋊C8, C4⋊Dic3 [×2], C4⋊Dic3, C4×C12, C2×C24, C2×Dic6, C4.10D8, C12⋊C8, C3×C4⋊C8, C122Q8, C12.2D8
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D6, C22⋊C4, D8, SD16 [×2], Q16, C4×S3, D12, C3⋊D4, C4.10D4, D4⋊C4, Q8⋊C4, C24⋊C2, D24, D6⋊C4, D4.S3, C3⋊Q16, C4.10D8, C6.SD16, C2.D24, C12.47D4, C12.2D8

Smallest permutation representation of C12.2D8
Regular action on 192 points
Generators in S192
```(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)(145 146 147 148 149 150 151 152 153 154 155 156)(157 158 159 160 161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190 191 192)
(1 38 125 112 181 59 27 66)(2 45 126 119 182 54 28 61)(3 40 127 114 183 49 29 68)(4 47 128 109 184 56 30 63)(5 42 129 116 185 51 31 70)(6 37 130 111 186 58 32 65)(7 44 131 118 187 53 33 72)(8 39 132 113 188 60 34 67)(9 46 121 120 189 55 35 62)(10 41 122 115 190 50 36 69)(11 48 123 110 191 57 25 64)(12 43 124 117 192 52 26 71)(13 169 135 104 154 88 165 84)(14 176 136 99 155 95 166 79)(15 171 137 106 156 90 167 74)(16 178 138 101 145 85 168 81)(17 173 139 108 146 92 157 76)(18 180 140 103 147 87 158 83)(19 175 141 98 148 94 159 78)(20 170 142 105 149 89 160 73)(21 177 143 100 150 96 161 80)(22 172 144 107 151 91 162 75)(23 179 133 102 152 86 163 82)(24 174 134 97 153 93 164 77)
(1 159 4 162 7 165 10 168)(2 164 5 167 8 158 11 161)(3 157 6 160 9 163 12 166)(13 36 16 27 19 30 22 33)(14 29 17 32 20 35 23 26)(15 34 18 25 21 28 24 31)(37 86 40 89 43 92 46 95)(38 91 41 94 44 85 47 88)(39 96 42 87 45 90 48 93)(49 170 52 173 55 176 58 179)(50 175 53 178 56 169 59 172)(51 180 54 171 57 174 60 177)(61 74 64 77 67 80 70 83)(62 79 65 82 68 73 71 76)(63 84 66 75 69 78 72 81)(97 113 100 116 103 119 106 110)(98 118 101 109 104 112 107 115)(99 111 102 114 105 117 108 120)(121 152 124 155 127 146 130 149)(122 145 125 148 128 151 131 154)(123 150 126 153 129 156 132 147)(133 192 136 183 139 186 142 189)(134 185 137 188 140 191 143 182)(135 190 138 181 141 184 144 187)```

`G:=sub<Sym(192)| (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)(145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192), (1,38,125,112,181,59,27,66)(2,45,126,119,182,54,28,61)(3,40,127,114,183,49,29,68)(4,47,128,109,184,56,30,63)(5,42,129,116,185,51,31,70)(6,37,130,111,186,58,32,65)(7,44,131,118,187,53,33,72)(8,39,132,113,188,60,34,67)(9,46,121,120,189,55,35,62)(10,41,122,115,190,50,36,69)(11,48,123,110,191,57,25,64)(12,43,124,117,192,52,26,71)(13,169,135,104,154,88,165,84)(14,176,136,99,155,95,166,79)(15,171,137,106,156,90,167,74)(16,178,138,101,145,85,168,81)(17,173,139,108,146,92,157,76)(18,180,140,103,147,87,158,83)(19,175,141,98,148,94,159,78)(20,170,142,105,149,89,160,73)(21,177,143,100,150,96,161,80)(22,172,144,107,151,91,162,75)(23,179,133,102,152,86,163,82)(24,174,134,97,153,93,164,77), (1,159,4,162,7,165,10,168)(2,164,5,167,8,158,11,161)(3,157,6,160,9,163,12,166)(13,36,16,27,19,30,22,33)(14,29,17,32,20,35,23,26)(15,34,18,25,21,28,24,31)(37,86,40,89,43,92,46,95)(38,91,41,94,44,85,47,88)(39,96,42,87,45,90,48,93)(49,170,52,173,55,176,58,179)(50,175,53,178,56,169,59,172)(51,180,54,171,57,174,60,177)(61,74,64,77,67,80,70,83)(62,79,65,82,68,73,71,76)(63,84,66,75,69,78,72,81)(97,113,100,116,103,119,106,110)(98,118,101,109,104,112,107,115)(99,111,102,114,105,117,108,120)(121,152,124,155,127,146,130,149)(122,145,125,148,128,151,131,154)(123,150,126,153,129,156,132,147)(133,192,136,183,139,186,142,189)(134,185,137,188,140,191,143,182)(135,190,138,181,141,184,144,187)>;`

`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)(145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192), (1,38,125,112,181,59,27,66)(2,45,126,119,182,54,28,61)(3,40,127,114,183,49,29,68)(4,47,128,109,184,56,30,63)(5,42,129,116,185,51,31,70)(6,37,130,111,186,58,32,65)(7,44,131,118,187,53,33,72)(8,39,132,113,188,60,34,67)(9,46,121,120,189,55,35,62)(10,41,122,115,190,50,36,69)(11,48,123,110,191,57,25,64)(12,43,124,117,192,52,26,71)(13,169,135,104,154,88,165,84)(14,176,136,99,155,95,166,79)(15,171,137,106,156,90,167,74)(16,178,138,101,145,85,168,81)(17,173,139,108,146,92,157,76)(18,180,140,103,147,87,158,83)(19,175,141,98,148,94,159,78)(20,170,142,105,149,89,160,73)(21,177,143,100,150,96,161,80)(22,172,144,107,151,91,162,75)(23,179,133,102,152,86,163,82)(24,174,134,97,153,93,164,77), (1,159,4,162,7,165,10,168)(2,164,5,167,8,158,11,161)(3,157,6,160,9,163,12,166)(13,36,16,27,19,30,22,33)(14,29,17,32,20,35,23,26)(15,34,18,25,21,28,24,31)(37,86,40,89,43,92,46,95)(38,91,41,94,44,85,47,88)(39,96,42,87,45,90,48,93)(49,170,52,173,55,176,58,179)(50,175,53,178,56,169,59,172)(51,180,54,171,57,174,60,177)(61,74,64,77,67,80,70,83)(62,79,65,82,68,73,71,76)(63,84,66,75,69,78,72,81)(97,113,100,116,103,119,106,110)(98,118,101,109,104,112,107,115)(99,111,102,114,105,117,108,120)(121,152,124,155,127,146,130,149)(122,145,125,148,128,151,131,154)(123,150,126,153,129,156,132,147)(133,192,136,183,139,186,142,189)(134,185,137,188,140,191,143,182)(135,190,138,181,141,184,144,187) );`

`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),(145,146,147,148,149,150,151,152,153,154,155,156),(157,158,159,160,161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176,177,178,179,180),(181,182,183,184,185,186,187,188,189,190,191,192)], [(1,38,125,112,181,59,27,66),(2,45,126,119,182,54,28,61),(3,40,127,114,183,49,29,68),(4,47,128,109,184,56,30,63),(5,42,129,116,185,51,31,70),(6,37,130,111,186,58,32,65),(7,44,131,118,187,53,33,72),(8,39,132,113,188,60,34,67),(9,46,121,120,189,55,35,62),(10,41,122,115,190,50,36,69),(11,48,123,110,191,57,25,64),(12,43,124,117,192,52,26,71),(13,169,135,104,154,88,165,84),(14,176,136,99,155,95,166,79),(15,171,137,106,156,90,167,74),(16,178,138,101,145,85,168,81),(17,173,139,108,146,92,157,76),(18,180,140,103,147,87,158,83),(19,175,141,98,148,94,159,78),(20,170,142,105,149,89,160,73),(21,177,143,100,150,96,161,80),(22,172,144,107,151,91,162,75),(23,179,133,102,152,86,163,82),(24,174,134,97,153,93,164,77)], [(1,159,4,162,7,165,10,168),(2,164,5,167,8,158,11,161),(3,157,6,160,9,163,12,166),(13,36,16,27,19,30,22,33),(14,29,17,32,20,35,23,26),(15,34,18,25,21,28,24,31),(37,86,40,89,43,92,46,95),(38,91,41,94,44,85,47,88),(39,96,42,87,45,90,48,93),(49,170,52,173,55,176,58,179),(50,175,53,178,56,169,59,172),(51,180,54,171,57,174,60,177),(61,74,64,77,67,80,70,83),(62,79,65,82,68,73,71,76),(63,84,66,75,69,78,72,81),(97,113,100,116,103,119,106,110),(98,118,101,109,104,112,107,115),(99,111,102,114,105,117,108,120),(121,152,124,155,127,146,130,149),(122,145,125,148,128,151,131,154),(123,150,126,153,129,156,132,147),(133,192,136,183,139,186,142,189),(134,185,137,188,140,191,143,182),(135,190,138,181,141,184,144,187)])`

39 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 6A 6B 6C 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 12C 12D 12E 12F 12G 12H 24A ··· 24H order 1 2 2 2 3 4 4 4 4 4 4 4 6 6 6 8 8 8 8 8 8 8 8 12 12 12 12 12 12 12 12 24 ··· 24 size 1 1 1 1 2 2 2 2 2 4 24 24 2 2 2 4 4 4 4 12 12 12 12 2 2 2 2 4 4 4 4 4 ··· 4

39 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + - + + - - - - image C1 C2 C2 C2 C4 S3 D4 D6 D8 SD16 Q16 C4×S3 D12 C3⋊D4 C24⋊C2 D24 C4.10D4 D4.S3 C3⋊Q16 C12.47D4 kernel C12.2D8 C12⋊C8 C3×C4⋊C8 C12⋊2Q8 C4⋊Dic3 C4⋊C8 C2×C12 C42 C12 C12 C12 C2×C4 C2×C4 C2×C4 C4 C4 C6 C4 C4 C2 # reps 1 1 1 1 4 1 2 1 2 4 2 2 2 2 4 4 1 1 1 2

Matrix representation of C12.2D8 in GL4(𝔽73) generated by

 65 0 0 0 44 9 0 0 0 0 72 2 0 0 72 1
,
 63 0 0 0 27 22 0 0 0 0 17 57 0 0 9 56
,
 69 25 0 0 11 4 0 0 0 0 41 32 0 0 57 0
`G:=sub<GL(4,GF(73))| [65,44,0,0,0,9,0,0,0,0,72,72,0,0,2,1],[63,27,0,0,0,22,0,0,0,0,17,9,0,0,57,56],[69,11,0,0,25,4,0,0,0,0,41,57,0,0,32,0] >;`

C12.2D8 in GAP, Magma, Sage, TeX

`C_{12}._2D_8`
`% in TeX`

`G:=Group("C12.2D8");`
`// GroupNames label`

`G:=SmallGroup(192,45);`
`// by ID`

`G=gap.SmallGroup(192,45);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,141,36,1094,268,1123,346,136,6278]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=b^8=1,c^2=a^3,b*a*b^-1=a^7,c*a*c^-1=a^5,c*b*c^-1=a^9*b^-1>;`
`// generators/relations`

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