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## G = (C2×C12).288D4order 192 = 26·3

### 262nd non-split extension by C2×C12 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — (C2×C12).288D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C2×Dic3⋊C4 — (C2×C12).288D4
 Lower central C3 — C22×C6 — (C2×C12).288D4
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for (C2×C12).288D4
G = < a,b,c,d | a2=b12=c4=1, d2=b6, ab=ba, ac=ca, ad=da, cbc-1=ab-1, dbd-1=ab5, dcd-1=ac-1 >

Subgroups: 328 in 134 conjugacy classes, 57 normal (25 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×9], C22 [×3], C22 [×4], C6 [×3], C6 [×4], C2×C4 [×2], C2×C4 [×21], C23, Dic3 [×5], C12 [×4], C2×C6 [×3], C2×C6 [×4], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×Dic3 [×2], C2×Dic3 [×11], C2×C12 [×2], C2×C12 [×8], C22×C6, C2.C42 [×5], C2×C4⋊C4, C2×C4⋊C4, Dic3⋊C4 [×2], C3×C4⋊C4 [×2], C22×Dic3 [×2], C22×Dic3 [×2], C22×C12, C22×C12 [×2], C23.83C23, C6.C42, C6.C42 [×4], C2×Dic3⋊C4, C6×C4⋊C4, (C2×C12).288D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D6 [×3], C2×D4, C2×Q8, C4○D4 [×5], C3⋊D4 [×2], C22×S3, C22⋊Q8, C22.D4, C4.4D4, C42.C2 [×2], C422C2 [×2], C4○D12 [×2], D42S3 [×2], S3×Q8, Q83S3, C2×C3⋊D4, C23.83C23, Dic3.Q8 [×2], C4⋊C4⋊S3 [×2], C23.28D6, C23.12D6, D63Q8, (C2×C12).288D4

Smallest permutation representation of (C2×C12).288D4
Regular action on 192 points
Generators in S192
(1 186)(2 187)(3 188)(4 189)(5 190)(6 191)(7 192)(8 181)(9 182)(10 183)(11 184)(12 185)(13 159)(14 160)(15 161)(16 162)(17 163)(18 164)(19 165)(20 166)(21 167)(22 168)(23 157)(24 158)(25 72)(26 61)(27 62)(28 63)(29 64)(30 65)(31 66)(32 67)(33 68)(34 69)(35 70)(36 71)(37 90)(38 91)(39 92)(40 93)(41 94)(42 95)(43 96)(44 85)(45 86)(46 87)(47 88)(48 89)(49 82)(50 83)(51 84)(52 73)(53 74)(54 75)(55 76)(56 77)(57 78)(58 79)(59 80)(60 81)(97 139)(98 140)(99 141)(100 142)(101 143)(102 144)(103 133)(104 134)(105 135)(106 136)(107 137)(108 138)(109 173)(110 174)(111 175)(112 176)(113 177)(114 178)(115 179)(116 180)(117 169)(118 170)(119 171)(120 172)(121 152)(122 153)(123 154)(124 155)(125 156)(126 145)(127 146)(128 147)(129 148)(130 149)(131 150)(132 151)
(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 57 159 177)(2 77 160 112)(3 55 161 175)(4 75 162 110)(5 53 163 173)(6 73 164 120)(7 51 165 171)(8 83 166 118)(9 49 167 169)(10 81 168 116)(11 59 157 179)(12 79 158 114)(13 113 186 78)(14 176 187 56)(15 111 188 76)(16 174 189 54)(17 109 190 74)(18 172 191 52)(19 119 192 84)(20 170 181 50)(21 117 182 82)(22 180 183 60)(23 115 184 80)(24 178 185 58)(25 126 140 39)(26 156 141 91)(27 124 142 37)(28 154 143 89)(29 122 144 47)(30 152 133 87)(31 132 134 45)(32 150 135 85)(33 130 136 43)(34 148 137 95)(35 128 138 41)(36 146 139 93)(38 61 125 99)(40 71 127 97)(42 69 129 107)(44 67 131 105)(46 65 121 103)(48 63 123 101)(62 155 100 90)(64 153 102 88)(66 151 104 86)(68 149 106 96)(70 147 108 94)(72 145 98 92)
(1 127 7 121)(2 151 8 145)(3 125 9 131)(4 149 10 155)(5 123 11 129)(6 147 12 153)(13 93 19 87)(14 45 20 39)(15 91 21 85)(16 43 22 37)(17 89 23 95)(18 41 24 47)(25 77 31 83)(26 49 32 55)(27 75 33 81)(28 59 34 53)(29 73 35 79)(30 57 36 51)(38 167 44 161)(40 165 46 159)(42 163 48 157)(50 72 56 66)(52 70 58 64)(54 68 60 62)(61 82 67 76)(63 80 69 74)(65 78 71 84)(86 166 92 160)(88 164 94 158)(90 162 96 168)(97 119 103 113)(98 176 104 170)(99 117 105 111)(100 174 106 180)(101 115 107 109)(102 172 108 178)(110 136 116 142)(112 134 118 140)(114 144 120 138)(122 191 128 185)(124 189 130 183)(126 187 132 181)(133 177 139 171)(135 175 141 169)(137 173 143 179)(146 192 152 186)(148 190 154 184)(150 188 156 182)

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

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

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

42 conjugacy classes

 class 1 2A ··· 2G 3 4A ··· 4F 4G ··· 4N 6A ··· 6G 12A ··· 12L order 1 2 ··· 2 3 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 4 ··· 4 12 ··· 12 2 ··· 2 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + - + + - - + image C1 C2 C2 C2 S3 Q8 D4 D6 C4○D4 C3⋊D4 C4○D12 D4⋊2S3 S3×Q8 Q8⋊3S3 kernel (C2×C12).288D4 C6.C42 C2×Dic3⋊C4 C6×C4⋊C4 C2×C4⋊C4 C2×Dic3 C2×C12 C22×C4 C2×C6 C2×C4 C22 C22 C22 C22 # reps 1 5 1 1 1 2 2 3 10 4 8 2 1 1

Matrix representation of (C2×C12).288D4 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 4 3 0 0 0 0 3 9 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 9 0 0 0 0 0 0 10
,
 10 4 0 0 0 0 4 3 0 0 0 0 0 0 5 0 0 0 0 0 0 8 0 0 0 0 0 0 0 5 0 0 0 0 5 0
,
 9 10 0 0 0 0 10 4 0 0 0 0 0 0 0 8 0 0 0 0 5 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[4,3,0,0,0,0,3,9,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,9,0,0,0,0,0,0,10],[10,4,0,0,0,0,4,3,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0,0,0,0,0,0,0,5,0,0,0,0,5,0],[9,10,0,0,0,0,10,4,0,0,0,0,0,0,0,5,0,0,0,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

(C2×C12).288D4 in GAP, Magma, Sage, TeX

(C_2\times C_{12})._{288}D_4
% in TeX

G:=Group("(C2xC12).288D4");
// GroupNames label

G:=SmallGroup(192,544);
// by ID

G=gap.SmallGroup(192,544);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,232,254,387,268,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^12=c^4=1,d^2=b^6,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^-1,d*b*d^-1=a*b^5,d*c*d^-1=a*c^-1>;
// generators/relations

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