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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, C3, C4, C22, C6, C2×C4, C2×C4, Q8, C23, Dic3, Dic3, C12, C2×C6, C42, C4⋊C4, C22×C4, C22×C4, C2×Q8, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C2.C42, C2.C42, C2×C42, C2×C4⋊C4, C22×Q8, C4×Dic3, Dic3⋊C4, C2×Dic6, C2×Dic6, C22×Dic3, C22×C12, C23.67C23, C6.C42, C3×C2.C42, C2×C4×Dic3, C2×Dic3⋊C4, C22×Dic6, (C2×C12)⋊Q8
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D6, C22⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic6, C4×S3, C22×S3, C2×C22⋊C4, C4×Q8, C22⋊Q8, C4.4D4, C4⋊Q8, C2×Dic6, S3×C2×C4, C4○D12, S3×D4, 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
(1 110)(2 111)(3 112)(4 113)(5 114)(6 115)(7 116)(8 117)(9 118)(10 119)(11 120)(12 109)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 48)(22 37)(23 38)(24 39)(25 162)(26 163)(27 164)(28 165)(29 166)(30 167)(31 168)(32 157)(33 158)(34 159)(35 160)(36 161)(49 87)(50 88)(51 89)(52 90)(53 91)(54 92)(55 93)(56 94)(57 95)(58 96)(59 85)(60 86)(61 154)(62 155)(63 156)(64 145)(65 146)(66 147)(67 148)(68 149)(69 150)(70 151)(71 152)(72 153)(73 174)(74 175)(75 176)(76 177)(77 178)(78 179)(79 180)(80 169)(81 170)(82 171)(83 172)(84 173)(97 128)(98 129)(99 130)(100 131)(101 132)(102 121)(103 122)(104 123)(105 124)(106 125)(107 126)(108 127)(133 185)(134 186)(135 187)(136 188)(137 189)(138 190)(139 191)(140 192)(141 181)(142 182)(143 183)(144 184)
(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 75 145 137)(2 177 146 190)(3 77 147 139)(4 179 148 192)(5 79 149 141)(6 169 150 182)(7 81 151 143)(8 171 152 184)(9 83 153 133)(10 173 154 186)(11 73 155 135)(12 175 156 188)(13 131 31 55)(14 101 32 94)(15 121 33 57)(16 103 34 96)(17 123 35 59)(18 105 36 86)(19 125 25 49)(20 107 26 88)(21 127 27 51)(22 97 28 90)(23 129 29 53)(24 99 30 92)(37 128 165 52)(38 98 166 91)(39 130 167 54)(40 100 168 93)(41 132 157 56)(42 102 158 95)(43 122 159 58)(44 104 160 85)(45 124 161 60)(46 106 162 87)(47 126 163 50)(48 108 164 89)(61 134 119 84)(62 187 120 174)(63 136 109 74)(64 189 110 176)(65 138 111 76)(66 191 112 178)(67 140 113 78)(68 181 114 180)(69 142 115 80)(70 183 116 170)(71 144 117 82)(72 185 118 172)
(1 34 145 16)(2 27 146 21)(3 32 147 14)(4 25 148 19)(5 30 149 24)(6 35 150 17)(7 28 151 22)(8 33 152 15)(9 26 153 20)(10 31 154 13)(11 36 155 18)(12 29 156 23)(37 116 165 70)(38 109 166 63)(39 114 167 68)(40 119 168 61)(41 112 157 66)(42 117 158 71)(43 110 159 64)(44 115 160 69)(45 120 161 62)(46 113 162 67)(47 118 163 72)(48 111 164 65)(49 179 125 192)(50 172 126 185)(51 177 127 190)(52 170 128 183)(53 175 129 188)(54 180 130 181)(55 173 131 186)(56 178 132 191)(57 171 121 184)(58 176 122 189)(59 169 123 182)(60 174 124 187)(73 105 135 86)(74 98 136 91)(75 103 137 96)(76 108 138 89)(77 101 139 94)(78 106 140 87)(79 99 141 92)(80 104 142 85)(81 97 143 90)(82 102 144 95)(83 107 133 88)(84 100 134 93)

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

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

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

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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