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## G = (C2×C12)⋊Q8order 192 = 26·3

### 6th semidirect product of C2×C12 and Q8 acting via Q8/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — (C2×C12)⋊Q8
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C22×Dic6 — (C2×C12)⋊Q8
 Lower central C3 — C2×C6 — (C2×C12)⋊Q8
 Upper central C1 — C23 — C2.C42

Generators and relations for (C2×C12)⋊Q8
G = < a,b,c,d | a2=b12=c4=1, d2=c2, cbc-1=ab=ba, ac=ca, ad=da, dbd-1=b5, dcd-1=c-1 >

Subgroups: 448 in 186 conjugacy classes, 77 normal (51 characteristic)
C1, C2 [×7], C3, C4 [×14], C22 [×7], C6 [×7], C2×C4 [×4], C2×C4 [×24], Q8 [×8], C23, Dic3 [×4], Dic3 [×5], C12 [×5], C2×C6 [×7], C42 [×2], C4⋊C4 [×2], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×8], Dic6 [×8], C2×Dic3 [×10], C2×Dic3 [×7], C2×C12 [×4], C2×C12 [×7], C22×C6, C2.C42, C2.C42 [×3], C2×C42, C2×C4⋊C4, C22×Q8, C4×Dic3 [×2], Dic3⋊C4 [×2], C2×Dic6 [×4], C2×Dic6 [×4], C22×Dic3 [×4], C22×C12 [×3], C23.67C23, C6.C42 [×3], C3×C2.C42, C2×C4×Dic3, C2×Dic3⋊C4, C22×Dic6, (C2×C12)⋊Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], Q8 [×4], C23, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], Dic6 [×2], C4×S3 [×2], C22×S3, C2×C22⋊C4, C4×Q8 [×2], C22⋊Q8 [×2], C4.4D4, C4⋊Q8, C2×Dic6, S3×C2×C4, C4○D12, S3×D4 [×2], D42S3, S3×Q8, C23.67C23, C4×Dic6, Dic3.D4, S3×C22⋊C4, C23.11D6, Dic6⋊C4, C12⋊Q8, D6⋊Q8, (C2×C12)⋊Q8

Smallest permutation representation of (C2×C12)⋊Q8
Regular action on 192 points
Generators in S192
(1 159)(2 160)(3 161)(4 162)(5 163)(6 164)(7 165)(8 166)(9 167)(10 168)(11 157)(12 158)(13 69)(14 70)(15 71)(16 72)(17 61)(18 62)(19 63)(20 64)(21 65)(22 66)(23 67)(24 68)(25 54)(26 55)(27 56)(28 57)(29 58)(30 59)(31 60)(32 49)(33 50)(34 51)(35 52)(36 53)(37 106)(38 107)(39 108)(40 97)(41 98)(42 99)(43 100)(44 101)(45 102)(46 103)(47 104)(48 105)(73 156)(74 145)(75 146)(76 147)(77 148)(78 149)(79 150)(80 151)(81 152)(82 153)(83 154)(84 155)(85 191)(86 192)(87 181)(88 182)(89 183)(90 184)(91 185)(92 186)(93 187)(94 188)(95 189)(96 190)(109 171)(110 172)(111 173)(112 174)(113 175)(114 176)(115 177)(116 178)(117 179)(118 180)(119 169)(120 170)(121 143)(122 144)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)(131 141)(132 142)
(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)
(1 177 140 78)(2 116 141 150)(3 179 142 80)(4 118 143 152)(5 169 144 82)(6 120 133 154)(7 171 134 84)(8 110 135 156)(9 173 136 74)(10 112 137 146)(11 175 138 76)(12 114 139 148)(13 93 52 99)(14 188 53 43)(15 95 54 101)(16 190 55 45)(17 85 56 103)(18 192 57 47)(19 87 58 105)(20 182 59 37)(21 89 60 107)(22 184 49 39)(23 91 50 97)(24 186 51 41)(25 44 71 189)(26 102 72 96)(27 46 61 191)(28 104 62 86)(29 48 63 181)(30 106 64 88)(31 38 65 183)(32 108 66 90)(33 40 67 185)(34 98 68 92)(35 42 69 187)(36 100 70 94)(73 166 172 125)(75 168 174 127)(77 158 176 129)(79 160 178 131)(81 162 180 121)(83 164 170 123)(109 124 155 165)(111 126 145 167)(113 128 147 157)(115 130 149 159)(117 132 151 161)(119 122 153 163)
(1 97 140 91)(2 102 141 96)(3 107 142 89)(4 100 143 94)(5 105 144 87)(6 98 133 92)(7 103 134 85)(8 108 135 90)(9 101 136 95)(10 106 137 88)(11 99 138 93)(12 104 139 86)(13 175 52 76)(14 180 53 81)(15 173 54 74)(16 178 55 79)(17 171 56 84)(18 176 57 77)(19 169 58 82)(20 174 59 75)(21 179 60 80)(22 172 49 73)(23 177 50 78)(24 170 51 83)(25 145 71 111)(26 150 72 116)(27 155 61 109)(28 148 62 114)(29 153 63 119)(30 146 64 112)(31 151 65 117)(32 156 66 110)(33 149 67 115)(34 154 68 120)(35 147 69 113)(36 152 70 118)(37 127 182 168)(38 132 183 161)(39 125 184 166)(40 130 185 159)(41 123 186 164)(42 128 187 157)(43 121 188 162)(44 126 189 167)(45 131 190 160)(46 124 191 165)(47 129 192 158)(48 122 181 163)

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

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

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

48 conjugacy classes

 class 1 2A ··· 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 4Q 4R 4S 4T 6A ··· 6G 12A ··· 12L order 1 2 ··· 2 3 4 4 4 4 4 4 4 4 4 ··· 4 4 4 4 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 2 2 2 2 4 4 4 4 6 ··· 6 12 12 12 12 2 ··· 2 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + - - + - + - - image C1 C2 C2 C2 C2 C2 C4 S3 D4 Q8 Q8 D6 C4○D4 Dic6 C4×S3 C4○D12 S3×D4 D4⋊2S3 S3×Q8 kernel (C2×C12)⋊Q8 C6.C42 C3×C2.C42 C2×C4×Dic3 C2×Dic3⋊C4 C22×Dic6 C2×Dic6 C2.C42 C2×Dic3 C2×Dic3 C2×C12 C22×C4 C2×C6 C2×C4 C2×C4 C22 C22 C22 C22 # reps 1 3 1 1 1 1 8 1 4 2 2 3 4 4 4 4 2 1 1

Matrix representation of (C2×C12)⋊Q8 in GL8(𝔽13)

 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 8 4
,
 6 2 0 0 0 0 0 0 2 7 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 3 2 0 0 0 0 0 0 8 10 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 2 5 0 0 0 0 0 0 0 0 6 2 0 0 0 0 0 0 2 7

G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,10,8,0,0,0,0,0,0,0,4],[6,2,0,0,0,0,0,0,2,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,3,8,0,0,0,0,0,0,2,10,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,8,2,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,6,2,0,0,0,0,0,0,2,7] >;

(C2×C12)⋊Q8 in GAP, Magma, Sage, TeX

(C_2\times C_{12})\rtimes Q_8
% in TeX

G:=Group("(C2xC12):Q8");
// GroupNames label

G:=SmallGroup(192,205);
// by ID

G=gap.SmallGroup(192,205);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,120,387,58,6278]);
// Polycyclic

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

׿
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