metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C12).1Q8, C2.7(C12⋊Q8), C6.12(C4⋊Q8), (C2×C4).8Dic6, C6.2(C4⋊D4), (C2×Dic3).8D4, (C2×Dic3).1Q8, (C22×C4).84D6, C2.8(Dic3⋊D4), C22.39(S3×Q8), C2.8(D6⋊Q8), C22.151(S3×D4), C6.21(C22⋊Q8), C2.8(C23.9D6), C2.8(Dic3.Q8), C23.32D6.5C2, (C22×C12).7C22, C6.11(C42.C2), C2.4(C12.6Q8), C22.41(C2×Dic6), C22.84(C4○D12), C2.C42.14S3, (C22×C6).282C23, C23.363(C22×S3), C6.4(C22.D4), C22.82(D4⋊2S3), C3⋊1(C23.81C23), C2.9(Dic3.D4), (C22×Dic3).7C22, (C2×C6).63(C2×Q8), (C2×C6).194(C2×D4), (C2×C4⋊Dic3).7C2, (C2×C6).57(C4○D4), (C2×Dic3⋊C4).6C2, (C3×C2.C42).9C2, SmallGroup(192,216)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2.(C12⋊Q8)
G = < a,b,c,d | a2=b12=c4=1, d2=c2, ab=ba, ac=ca, ad=da, cbc-1=ab7, dbd-1=ab5, dcd-1=ac-1 >
Subgroups: 368 in 150 conjugacy classes, 61 normal (51 characteristic)
C1, C2 [×7], C3, C4 [×11], C22 [×7], C6 [×7], C2×C4 [×2], C2×C4 [×23], C23, Dic3 [×7], C12 [×4], C2×C6 [×7], C4⋊C4 [×8], C22×C4 [×3], C22×C4 [×4], C2×Dic3 [×6], C2×Dic3 [×9], C2×C12 [×2], C2×C12 [×8], C22×C6, C2.C42, C2.C42 [×2], C2×C4⋊C4 [×4], Dic3⋊C4 [×6], C4⋊Dic3 [×2], C22×Dic3 [×4], C22×C12 [×3], C23.81C23, C23.32D6 [×2], C3×C2.C42, C2×Dic3⋊C4 [×3], C2×C4⋊Dic3, C2.(C12⋊Q8)
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], Q8 [×4], C23, D6 [×3], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×3], Dic6 [×2], C22×S3, C4⋊D4, C22⋊Q8 [×2], C22.D4, C42.C2 [×2], C4⋊Q8, C2×Dic6, C4○D12 [×2], S3×D4 [×2], D4⋊2S3, S3×Q8, C23.81C23, C12.6Q8, Dic3.D4, C23.9D6, Dic3⋊D4, C12⋊Q8, Dic3.Q8, D6⋊Q8, C2.(C12⋊Q8)
►Regular action on 192 points
Generators in S192
(1 94)(2 95)(3 96)(4 85)(5 86)(6 87)(7 88)(8 89)(9 90)(10 91)(11 92)(12 93)(13 35)(14 36)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 33)(24 34)(37 68)(38 69)(39 70)(40 71)(41 72)(42 61)(43 62)(44 63)(45 64)(46 65)(47 66)(48 67)(49 115)(50 116)(51 117)(52 118)(53 119)(54 120)(55 109)(56 110)(57 111)(58 112)(59 113)(60 114)(73 106)(74 107)(75 108)(76 97)(77 98)(78 99)(79 100)(80 101)(81 102)(82 103)(83 104)(84 105)(121 182)(122 183)(123 184)(124 185)(125 186)(126 187)(127 188)(128 189)(129 190)(130 191)(131 192)(132 181)(133 159)(134 160)(135 161)(136 162)(137 163)(138 164)(139 165)(140 166)(141 167)(142 168)(143 157)(144 158)(145 173)(146 174)(147 175)(148 176)(149 177)(150 178)(151 179)(152 180)(153 169)(154 170)(155 171)(156 172)
(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)
(1 153 137 112)(2 176 138 53)(3 155 139 114)(4 178 140 55)(5 145 141 116)(6 180 142 57)(7 147 143 118)(8 170 144 59)(9 149 133 120)(10 172 134 49)(11 151 135 110)(12 174 136 51)(13 76 183 43)(14 104 184 69)(15 78 185 45)(16 106 186 71)(17 80 187 47)(18 108 188 61)(19 82 189 37)(20 98 190 63)(21 84 191 39)(22 100 192 65)(23 74 181 41)(24 102 182 67)(25 99 124 64)(26 73 125 40)(27 101 126 66)(28 75 127 42)(29 103 128 68)(30 77 129 44)(31 105 130 70)(32 79 131 46)(33 107 132 72)(34 81 121 48)(35 97 122 62)(36 83 123 38)(50 86 173 167)(52 88 175 157)(54 90 177 159)(56 92 179 161)(58 94 169 163)(60 96 171 165)(85 150 166 109)(87 152 168 111)(89 154 158 113)(91 156 160 115)(93 146 162 117)(95 148 164 119)
(1 36 137 123)(2 19 138 189)(3 34 139 121)(4 17 140 187)(5 32 141 131)(6 15 142 185)(7 30 143 129)(8 13 144 183)(9 28 133 127)(10 23 134 181)(11 26 135 125)(12 21 136 191)(14 163 184 94)(16 161 186 92)(18 159 188 90)(20 157 190 88)(22 167 192 86)(24 165 182 96)(25 168 124 87)(27 166 126 85)(29 164 128 95)(31 162 130 93)(33 160 132 91)(35 158 122 89)(37 119 82 148)(38 58 83 169)(39 117 84 146)(40 56 73 179)(41 115 74 156)(42 54 75 177)(43 113 76 154)(44 52 77 175)(45 111 78 152)(46 50 79 173)(47 109 80 150)(48 60 81 171)(49 107 172 72)(51 105 174 70)(53 103 176 68)(55 101 178 66)(57 99 180 64)(59 97 170 62)(61 120 108 149)(63 118 98 147)(65 116 100 145)(67 114 102 155)(69 112 104 153)(71 110 106 151)
G:=sub<Sym(192)| (1,94)(2,95)(3,96)(4,85)(5,86)(6,87)(7,88)(8,89)(9,90)(10,91)(11,92)(12,93)(13,35)(14,36)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(21,31)(22,32)(23,33)(24,34)(37,68)(38,69)(39,70)(40,71)(41,72)(42,61)(43,62)(44,63)(45,64)(46,65)(47,66)(48,67)(49,115)(50,116)(51,117)(52,118)(53,119)(54,120)(55,109)(56,110)(57,111)(58,112)(59,113)(60,114)(73,106)(74,107)(75,108)(76,97)(77,98)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105)(121,182)(122,183)(123,184)(124,185)(125,186)(126,187)(127,188)(128,189)(129,190)(130,191)(131,192)(132,181)(133,159)(134,160)(135,161)(136,162)(137,163)(138,164)(139,165)(140,166)(141,167)(142,168)(143,157)(144,158)(145,173)(146,174)(147,175)(148,176)(149,177)(150,178)(151,179)(152,180)(153,169)(154,170)(155,171)(156,172), (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), (1,153,137,112)(2,176,138,53)(3,155,139,114)(4,178,140,55)(5,145,141,116)(6,180,142,57)(7,147,143,118)(8,170,144,59)(9,149,133,120)(10,172,134,49)(11,151,135,110)(12,174,136,51)(13,76,183,43)(14,104,184,69)(15,78,185,45)(16,106,186,71)(17,80,187,47)(18,108,188,61)(19,82,189,37)(20,98,190,63)(21,84,191,39)(22,100,192,65)(23,74,181,41)(24,102,182,67)(25,99,124,64)(26,73,125,40)(27,101,126,66)(28,75,127,42)(29,103,128,68)(30,77,129,44)(31,105,130,70)(32,79,131,46)(33,107,132,72)(34,81,121,48)(35,97,122,62)(36,83,123,38)(50,86,173,167)(52,88,175,157)(54,90,177,159)(56,92,179,161)(58,94,169,163)(60,96,171,165)(85,150,166,109)(87,152,168,111)(89,154,158,113)(91,156,160,115)(93,146,162,117)(95,148,164,119), (1,36,137,123)(2,19,138,189)(3,34,139,121)(4,17,140,187)(5,32,141,131)(6,15,142,185)(7,30,143,129)(8,13,144,183)(9,28,133,127)(10,23,134,181)(11,26,135,125)(12,21,136,191)(14,163,184,94)(16,161,186,92)(18,159,188,90)(20,157,190,88)(22,167,192,86)(24,165,182,96)(25,168,124,87)(27,166,126,85)(29,164,128,95)(31,162,130,93)(33,160,132,91)(35,158,122,89)(37,119,82,148)(38,58,83,169)(39,117,84,146)(40,56,73,179)(41,115,74,156)(42,54,75,177)(43,113,76,154)(44,52,77,175)(45,111,78,152)(46,50,79,173)(47,109,80,150)(48,60,81,171)(49,107,172,72)(51,105,174,70)(53,103,176,68)(55,101,178,66)(57,99,180,64)(59,97,170,62)(61,120,108,149)(63,118,98,147)(65,116,100,145)(67,114,102,155)(69,112,104,153)(71,110,106,151)>;
G:=Group( (1,94)(2,95)(3,96)(4,85)(5,86)(6,87)(7,88)(8,89)(9,90)(10,91)(11,92)(12,93)(13,35)(14,36)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(21,31)(22,32)(23,33)(24,34)(37,68)(38,69)(39,70)(40,71)(41,72)(42,61)(43,62)(44,63)(45,64)(46,65)(47,66)(48,67)(49,115)(50,116)(51,117)(52,118)(53,119)(54,120)(55,109)(56,110)(57,111)(58,112)(59,113)(60,114)(73,106)(74,107)(75,108)(76,97)(77,98)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105)(121,182)(122,183)(123,184)(124,185)(125,186)(126,187)(127,188)(128,189)(129,190)(130,191)(131,192)(132,181)(133,159)(134,160)(135,161)(136,162)(137,163)(138,164)(139,165)(140,166)(141,167)(142,168)(143,157)(144,158)(145,173)(146,174)(147,175)(148,176)(149,177)(150,178)(151,179)(152,180)(153,169)(154,170)(155,171)(156,172), (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), (1,153,137,112)(2,176,138,53)(3,155,139,114)(4,178,140,55)(5,145,141,116)(6,180,142,57)(7,147,143,118)(8,170,144,59)(9,149,133,120)(10,172,134,49)(11,151,135,110)(12,174,136,51)(13,76,183,43)(14,104,184,69)(15,78,185,45)(16,106,186,71)(17,80,187,47)(18,108,188,61)(19,82,189,37)(20,98,190,63)(21,84,191,39)(22,100,192,65)(23,74,181,41)(24,102,182,67)(25,99,124,64)(26,73,125,40)(27,101,126,66)(28,75,127,42)(29,103,128,68)(30,77,129,44)(31,105,130,70)(32,79,131,46)(33,107,132,72)(34,81,121,48)(35,97,122,62)(36,83,123,38)(50,86,173,167)(52,88,175,157)(54,90,177,159)(56,92,179,161)(58,94,169,163)(60,96,171,165)(85,150,166,109)(87,152,168,111)(89,154,158,113)(91,156,160,115)(93,146,162,117)(95,148,164,119), (1,36,137,123)(2,19,138,189)(3,34,139,121)(4,17,140,187)(5,32,141,131)(6,15,142,185)(7,30,143,129)(8,13,144,183)(9,28,133,127)(10,23,134,181)(11,26,135,125)(12,21,136,191)(14,163,184,94)(16,161,186,92)(18,159,188,90)(20,157,190,88)(22,167,192,86)(24,165,182,96)(25,168,124,87)(27,166,126,85)(29,164,128,95)(31,162,130,93)(33,160,132,91)(35,158,122,89)(37,119,82,148)(38,58,83,169)(39,117,84,146)(40,56,73,179)(41,115,74,156)(42,54,75,177)(43,113,76,154)(44,52,77,175)(45,111,78,152)(46,50,79,173)(47,109,80,150)(48,60,81,171)(49,107,172,72)(51,105,174,70)(53,103,176,68)(55,101,178,66)(57,99,180,64)(59,97,170,62)(61,120,108,149)(63,118,98,147)(65,116,100,145)(67,114,102,155)(69,112,104,153)(71,110,106,151) );
G=PermutationGroup([(1,94),(2,95),(3,96),(4,85),(5,86),(6,87),(7,88),(8,89),(9,90),(10,91),(11,92),(12,93),(13,35),(14,36),(15,25),(16,26),(17,27),(18,28),(19,29),(20,30),(21,31),(22,32),(23,33),(24,34),(37,68),(38,69),(39,70),(40,71),(41,72),(42,61),(43,62),(44,63),(45,64),(46,65),(47,66),(48,67),(49,115),(50,116),(51,117),(52,118),(53,119),(54,120),(55,109),(56,110),(57,111),(58,112),(59,113),(60,114),(73,106),(74,107),(75,108),(76,97),(77,98),(78,99),(79,100),(80,101),(81,102),(82,103),(83,104),(84,105),(121,182),(122,183),(123,184),(124,185),(125,186),(126,187),(127,188),(128,189),(129,190),(130,191),(131,192),(132,181),(133,159),(134,160),(135,161),(136,162),(137,163),(138,164),(139,165),(140,166),(141,167),(142,168),(143,157),(144,158),(145,173),(146,174),(147,175),(148,176),(149,177),(150,178),(151,179),(152,180),(153,169),(154,170),(155,171),(156,172)], [(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)], [(1,153,137,112),(2,176,138,53),(3,155,139,114),(4,178,140,55),(5,145,141,116),(6,180,142,57),(7,147,143,118),(8,170,144,59),(9,149,133,120),(10,172,134,49),(11,151,135,110),(12,174,136,51),(13,76,183,43),(14,104,184,69),(15,78,185,45),(16,106,186,71),(17,80,187,47),(18,108,188,61),(19,82,189,37),(20,98,190,63),(21,84,191,39),(22,100,192,65),(23,74,181,41),(24,102,182,67),(25,99,124,64),(26,73,125,40),(27,101,126,66),(28,75,127,42),(29,103,128,68),(30,77,129,44),(31,105,130,70),(32,79,131,46),(33,107,132,72),(34,81,121,48),(35,97,122,62),(36,83,123,38),(50,86,173,167),(52,88,175,157),(54,90,177,159),(56,92,179,161),(58,94,169,163),(60,96,171,165),(85,150,166,109),(87,152,168,111),(89,154,158,113),(91,156,160,115),(93,146,162,117),(95,148,164,119)], [(1,36,137,123),(2,19,138,189),(3,34,139,121),(4,17,140,187),(5,32,141,131),(6,15,142,185),(7,30,143,129),(8,13,144,183),(9,28,133,127),(10,23,134,181),(11,26,135,125),(12,21,136,191),(14,163,184,94),(16,161,186,92),(18,159,188,90),(20,157,190,88),(22,167,192,86),(24,165,182,96),(25,168,124,87),(27,166,126,85),(29,164,128,95),(31,162,130,93),(33,160,132,91),(35,158,122,89),(37,119,82,148),(38,58,83,169),(39,117,84,146),(40,56,73,179),(41,115,74,156),(42,54,75,177),(43,113,76,154),(44,52,77,175),(45,111,78,152),(46,50,79,173),(47,109,80,150),(48,60,81,171),(49,107,172,72),(51,105,174,70),(53,103,176,68),(55,101,178,66),(57,99,180,64),(59,97,170,62),(61,120,108,149),(63,118,98,147),(65,116,100,145),(67,114,102,155),(69,112,104,153),(71,110,106,151)])
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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | - | + | - | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | Q8 | D6 | C4○D4 | Dic6 | C4○D12 | S3×D4 | D4⋊2S3 | S3×Q8 |
| kernel | C2.(C12⋊Q8) | C23.32D6 | C3×C2.C42 | C2×Dic3⋊C4 | C2×C4⋊Dic3 | C2.C42 | C2×Dic3 | C2×Dic3 | C2×C12 | C22×C4 | C2×C6 | C2×C4 | C22 | C22 | C22 | C22 |
| # reps | 1 | 2 | 1 | 3 | 1 | 1 | 4 | 2 | 2 | 3 | 6 | 4 | 8 | 2 | 1 | 1 |
Matrix representation of C2.(C12⋊Q8) ►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 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 8 |
| 0 | 0 | 0 | 0 | 11 | 2 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 7 | 0 | 0 |
| 0 | 0 | 6 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 11 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 2 | 0 | 0 |
| 0 | 0 | 4 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 6 |
| 0 | 0 | 0 | 0 | 3 | 10 |
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],[0,8,0,0,0,0,8,0,0,0,0,0,0,0,1,12,0,0,0,0,1,0,0,0,0,0,0,0,11,11,0,0,0,0,8,2],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,10,6,0,0,0,0,7,3,0,0,0,0,0,0,1,0,0,0,0,0,11,12],[0,5,0,0,0,0,5,0,0,0,0,0,0,0,11,4,0,0,0,0,2,2,0,0,0,0,0,0,3,3,0,0,0,0,6,10] >;
C2.(C12⋊Q8) in GAP, Magma, Sage, TeX
C_2.(C_{12}\rtimes Q_8) % in TeX
G:=Group("C2.(C12:Q8)"); // GroupNames label
G:=SmallGroup(192,216);
// by ID
G=gap.SmallGroup(192,216);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,64,254,387,100,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^7,d*b*d^-1=a*b^5,d*c*d^-1=a*c^-1>;
// generators/relations