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## G = C12.47D8order 192 = 26·3

### 1st non-split extension by C12 of D8 acting via D8/D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C12.47D8
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C4×C12 — C12⋊C8 — C12.47D8
 Lower central C3 — C2×C6 — C2×C12 — C12.47D8
 Upper central C1 — C22 — C42 — C4⋊C8

Generators and relations for C12.47D8
G = < a,b,c | a12=b8=1, c2=a6, bab-1=cac-1=a-1, 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, C12.47D8
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, Dic12, D6⋊C4, D4⋊S3, Q82S3, C4.10D8, C6.D8, C2.Dic12, C12.47D4, C12.47D8

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

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

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

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

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 Dic12 C4.10D4 D4⋊S3 Q8⋊2S3 C12.47D4 kernel C12.47D8 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 C12.47D8 in GL4(𝔽73) generated by

 59 66 0 0 7 66 0 0 0 0 1 0 0 0 0 1
,
 10 51 0 0 41 63 0 0 0 0 0 38 0 0 48 32
,
 8 47 0 0 39 65 0 0 0 0 71 52 0 0 21 2
`G:=sub<GL(4,GF(73))| [59,7,0,0,66,66,0,0,0,0,1,0,0,0,0,1],[10,41,0,0,51,63,0,0,0,0,0,48,0,0,38,32],[8,39,0,0,47,65,0,0,0,0,71,21,0,0,52,2] >;`

C12.47D8 in GAP, Magma, Sage, TeX

`C_{12}._{47}D_8`
`% in TeX`

`G:=Group("C12.47D8");`
`// GroupNames label`

`G:=SmallGroup(192,41);`
`// by ID`

`G=gap.SmallGroup(192,41);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,85,316,422,387,268,570,136,6278]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=b^8=1,c^2=a^6,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^9*b^-1>;`
`// generators/relations`

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