metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.9(C4×Q8), (C2×C12)⋊10Q8, (C2×C4)⋊8Dic6, C6.7(C4⋊Q8), (C2×Dic6)⋊7C4, (C2×C4).89D12, C4.22(D6⋊C4), (C2×C12).470D4, (C2×C42).15S3, C2.11(C4×Dic6), C2.2(C12⋊2Q8), C22.34(C2×D12), (C22×C4).411D6, C6.53(C22⋊Q8), C12.46(C22⋊C4), C2.1(C42⋊7S3), C6.10(C4.4D4), (C22×Dic6).4C2, C22.17(C2×Dic6), C2.2(C12.48D4), C22.40(C4○D12), C6.C42.10C2, C23.272(C22×S3), (C22×C6).304C23, (C22×C12).470C22, C3⋊2(C23.67C23), (C22×Dic3).27C22, (C2×C4×C12).10C2, C2.5(C2×D6⋊C4), (C2×C6).24(C2×Q8), (C2×C4).109(C4×S3), (C2×C6).424(C2×D4), C6.31(C2×C22⋊C4), C22.117(S3×C2×C4), (C2×C12).224(C2×C4), (C2×C6).65(C4○D4), (C2×C4⋊Dic3).14C2, (C2×C6).94(C22×C4), C22.39(C2×C3⋊D4), (C2×C4).238(C3⋊D4), (C2×Dic3).25(C2×C4), SmallGroup(192,488)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×Dic6)⋊7C4
G = < a,b,c,d | a2=b12=d4=1, c2=b6, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=ab6c >
Subgroups: 440 in 186 conjugacy classes, 87 normal (23 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C42, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C2.C42, C2×C42, C2×C4⋊C4, C22×Q8, C4⋊Dic3, C4×C12, C2×Dic6, C2×Dic6, C22×Dic3, C22×C12, C22×C12, C23.67C23, C6.C42, C2×C4⋊Dic3, C2×C4×C12, C22×Dic6, (C2×Dic6)⋊7C4
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, D12, C3⋊D4, C22×S3, C2×C22⋊C4, C4×Q8, C22⋊Q8, C4.4D4, C4⋊Q8, D6⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C2×C3⋊D4, C23.67C23, C4×Dic6, C12⋊2Q8, C42⋊7S3, C12.48D4, C2×D6⋊C4, (C2×Dic6)⋊7C4
►Regular action on 192 points
Generators in S192
(1 61)(2 62)(3 63)(4 64)(5 65)(6 66)(7 67)(8 68)(9 69)(10 70)(11 71)(12 72)(13 123)(14 124)(15 125)(16 126)(17 127)(18 128)(19 129)(20 130)(21 131)(22 132)(23 121)(24 122)(25 60)(26 49)(27 50)(28 51)(29 52)(30 53)(31 54)(32 55)(33 56)(34 57)(35 58)(36 59)(37 95)(38 96)(39 85)(40 86)(41 87)(42 88)(43 89)(44 90)(45 91)(46 92)(47 93)(48 94)(73 191)(74 192)(75 181)(76 182)(77 183)(78 184)(79 185)(80 186)(81 187)(82 188)(83 189)(84 190)(97 159)(98 160)(99 161)(100 162)(101 163)(102 164)(103 165)(104 166)(105 167)(106 168)(107 157)(108 158)(109 138)(110 139)(111 140)(112 141)(113 142)(114 143)(115 144)(116 133)(117 134)(118 135)(119 136)(120 137)(145 172)(146 173)(147 174)(148 175)(149 176)(150 177)(151 178)(152 179)(153 180)(154 169)(155 170)(156 171)
(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)
(1 164 7 158)(2 163 8 157)(3 162 9 168)(4 161 10 167)(5 160 11 166)(6 159 12 165)(13 180 19 174)(14 179 20 173)(15 178 21 172)(16 177 22 171)(17 176 23 170)(18 175 24 169)(25 143 31 137)(26 142 32 136)(27 141 33 135)(28 140 34 134)(29 139 35 133)(30 138 36 144)(37 187 43 181)(38 186 44 192)(39 185 45 191)(40 184 46 190)(41 183 47 189)(42 182 48 188)(49 113 55 119)(50 112 56 118)(51 111 57 117)(52 110 58 116)(53 109 59 115)(54 120 60 114)(61 102 67 108)(62 101 68 107)(63 100 69 106)(64 99 70 105)(65 98 71 104)(66 97 72 103)(73 85 79 91)(74 96 80 90)(75 95 81 89)(76 94 82 88)(77 93 83 87)(78 92 84 86)(121 155 127 149)(122 154 128 148)(123 153 129 147)(124 152 130 146)(125 151 131 145)(126 150 132 156)
(1 144 38 172)(2 133 39 173)(3 134 40 174)(4 135 41 175)(5 136 42 176)(6 137 43 177)(7 138 44 178)(8 139 45 179)(9 140 46 180)(10 141 47 169)(11 142 48 170)(12 143 37 171)(13 106 28 84)(14 107 29 73)(15 108 30 74)(16 97 31 75)(17 98 32 76)(18 99 33 77)(19 100 34 78)(20 101 35 79)(21 102 36 80)(22 103 25 81)(23 104 26 82)(24 105 27 83)(49 188 121 166)(50 189 122 167)(51 190 123 168)(52 191 124 157)(53 192 125 158)(54 181 126 159)(55 182 127 160)(56 183 128 161)(57 184 129 162)(58 185 130 163)(59 186 131 164)(60 187 132 165)(61 115 96 145)(62 116 85 146)(63 117 86 147)(64 118 87 148)(65 119 88 149)(66 120 89 150)(67 109 90 151)(68 110 91 152)(69 111 92 153)(70 112 93 154)(71 113 94 155)(72 114 95 156)
G:=sub<Sym(192)| (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,123)(14,124)(15,125)(16,126)(17,127)(18,128)(19,129)(20,130)(21,131)(22,132)(23,121)(24,122)(25,60)(26,49)(27,50)(28,51)(29,52)(30,53)(31,54)(32,55)(33,56)(34,57)(35,58)(36,59)(37,95)(38,96)(39,85)(40,86)(41,87)(42,88)(43,89)(44,90)(45,91)(46,92)(47,93)(48,94)(73,191)(74,192)(75,181)(76,182)(77,183)(78,184)(79,185)(80,186)(81,187)(82,188)(83,189)(84,190)(97,159)(98,160)(99,161)(100,162)(101,163)(102,164)(103,165)(104,166)(105,167)(106,168)(107,157)(108,158)(109,138)(110,139)(111,140)(112,141)(113,142)(114,143)(115,144)(116,133)(117,134)(118,135)(119,136)(120,137)(145,172)(146,173)(147,174)(148,175)(149,176)(150,177)(151,178)(152,179)(153,180)(154,169)(155,170)(156,171), (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), (1,164,7,158)(2,163,8,157)(3,162,9,168)(4,161,10,167)(5,160,11,166)(6,159,12,165)(13,180,19,174)(14,179,20,173)(15,178,21,172)(16,177,22,171)(17,176,23,170)(18,175,24,169)(25,143,31,137)(26,142,32,136)(27,141,33,135)(28,140,34,134)(29,139,35,133)(30,138,36,144)(37,187,43,181)(38,186,44,192)(39,185,45,191)(40,184,46,190)(41,183,47,189)(42,182,48,188)(49,113,55,119)(50,112,56,118)(51,111,57,117)(52,110,58,116)(53,109,59,115)(54,120,60,114)(61,102,67,108)(62,101,68,107)(63,100,69,106)(64,99,70,105)(65,98,71,104)(66,97,72,103)(73,85,79,91)(74,96,80,90)(75,95,81,89)(76,94,82,88)(77,93,83,87)(78,92,84,86)(121,155,127,149)(122,154,128,148)(123,153,129,147)(124,152,130,146)(125,151,131,145)(126,150,132,156), (1,144,38,172)(2,133,39,173)(3,134,40,174)(4,135,41,175)(5,136,42,176)(6,137,43,177)(7,138,44,178)(8,139,45,179)(9,140,46,180)(10,141,47,169)(11,142,48,170)(12,143,37,171)(13,106,28,84)(14,107,29,73)(15,108,30,74)(16,97,31,75)(17,98,32,76)(18,99,33,77)(19,100,34,78)(20,101,35,79)(21,102,36,80)(22,103,25,81)(23,104,26,82)(24,105,27,83)(49,188,121,166)(50,189,122,167)(51,190,123,168)(52,191,124,157)(53,192,125,158)(54,181,126,159)(55,182,127,160)(56,183,128,161)(57,184,129,162)(58,185,130,163)(59,186,131,164)(60,187,132,165)(61,115,96,145)(62,116,85,146)(63,117,86,147)(64,118,87,148)(65,119,88,149)(66,120,89,150)(67,109,90,151)(68,110,91,152)(69,111,92,153)(70,112,93,154)(71,113,94,155)(72,114,95,156)>;
G:=Group( (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,123)(14,124)(15,125)(16,126)(17,127)(18,128)(19,129)(20,130)(21,131)(22,132)(23,121)(24,122)(25,60)(26,49)(27,50)(28,51)(29,52)(30,53)(31,54)(32,55)(33,56)(34,57)(35,58)(36,59)(37,95)(38,96)(39,85)(40,86)(41,87)(42,88)(43,89)(44,90)(45,91)(46,92)(47,93)(48,94)(73,191)(74,192)(75,181)(76,182)(77,183)(78,184)(79,185)(80,186)(81,187)(82,188)(83,189)(84,190)(97,159)(98,160)(99,161)(100,162)(101,163)(102,164)(103,165)(104,166)(105,167)(106,168)(107,157)(108,158)(109,138)(110,139)(111,140)(112,141)(113,142)(114,143)(115,144)(116,133)(117,134)(118,135)(119,136)(120,137)(145,172)(146,173)(147,174)(148,175)(149,176)(150,177)(151,178)(152,179)(153,180)(154,169)(155,170)(156,171), (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), (1,164,7,158)(2,163,8,157)(3,162,9,168)(4,161,10,167)(5,160,11,166)(6,159,12,165)(13,180,19,174)(14,179,20,173)(15,178,21,172)(16,177,22,171)(17,176,23,170)(18,175,24,169)(25,143,31,137)(26,142,32,136)(27,141,33,135)(28,140,34,134)(29,139,35,133)(30,138,36,144)(37,187,43,181)(38,186,44,192)(39,185,45,191)(40,184,46,190)(41,183,47,189)(42,182,48,188)(49,113,55,119)(50,112,56,118)(51,111,57,117)(52,110,58,116)(53,109,59,115)(54,120,60,114)(61,102,67,108)(62,101,68,107)(63,100,69,106)(64,99,70,105)(65,98,71,104)(66,97,72,103)(73,85,79,91)(74,96,80,90)(75,95,81,89)(76,94,82,88)(77,93,83,87)(78,92,84,86)(121,155,127,149)(122,154,128,148)(123,153,129,147)(124,152,130,146)(125,151,131,145)(126,150,132,156), (1,144,38,172)(2,133,39,173)(3,134,40,174)(4,135,41,175)(5,136,42,176)(6,137,43,177)(7,138,44,178)(8,139,45,179)(9,140,46,180)(10,141,47,169)(11,142,48,170)(12,143,37,171)(13,106,28,84)(14,107,29,73)(15,108,30,74)(16,97,31,75)(17,98,32,76)(18,99,33,77)(19,100,34,78)(20,101,35,79)(21,102,36,80)(22,103,25,81)(23,104,26,82)(24,105,27,83)(49,188,121,166)(50,189,122,167)(51,190,123,168)(52,191,124,157)(53,192,125,158)(54,181,126,159)(55,182,127,160)(56,183,128,161)(57,184,129,162)(58,185,130,163)(59,186,131,164)(60,187,132,165)(61,115,96,145)(62,116,85,146)(63,117,86,147)(64,118,87,148)(65,119,88,149)(66,120,89,150)(67,109,90,151)(68,110,91,152)(69,111,92,153)(70,112,93,154)(71,113,94,155)(72,114,95,156) );
G=PermutationGroup([[(1,61),(2,62),(3,63),(4,64),(5,65),(6,66),(7,67),(8,68),(9,69),(10,70),(11,71),(12,72),(13,123),(14,124),(15,125),(16,126),(17,127),(18,128),(19,129),(20,130),(21,131),(22,132),(23,121),(24,122),(25,60),(26,49),(27,50),(28,51),(29,52),(30,53),(31,54),(32,55),(33,56),(34,57),(35,58),(36,59),(37,95),(38,96),(39,85),(40,86),(41,87),(42,88),(43,89),(44,90),(45,91),(46,92),(47,93),(48,94),(73,191),(74,192),(75,181),(76,182),(77,183),(78,184),(79,185),(80,186),(81,187),(82,188),(83,189),(84,190),(97,159),(98,160),(99,161),(100,162),(101,163),(102,164),(103,165),(104,166),(105,167),(106,168),(107,157),(108,158),(109,138),(110,139),(111,140),(112,141),(113,142),(114,143),(115,144),(116,133),(117,134),(118,135),(119,136),(120,137),(145,172),(146,173),(147,174),(148,175),(149,176),(150,177),(151,178),(152,179),(153,180),(154,169),(155,170),(156,171)], [(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)], [(1,164,7,158),(2,163,8,157),(3,162,9,168),(4,161,10,167),(5,160,11,166),(6,159,12,165),(13,180,19,174),(14,179,20,173),(15,178,21,172),(16,177,22,171),(17,176,23,170),(18,175,24,169),(25,143,31,137),(26,142,32,136),(27,141,33,135),(28,140,34,134),(29,139,35,133),(30,138,36,144),(37,187,43,181),(38,186,44,192),(39,185,45,191),(40,184,46,190),(41,183,47,189),(42,182,48,188),(49,113,55,119),(50,112,56,118),(51,111,57,117),(52,110,58,116),(53,109,59,115),(54,120,60,114),(61,102,67,108),(62,101,68,107),(63,100,69,106),(64,99,70,105),(65,98,71,104),(66,97,72,103),(73,85,79,91),(74,96,80,90),(75,95,81,89),(76,94,82,88),(77,93,83,87),(78,92,84,86),(121,155,127,149),(122,154,128,148),(123,153,129,147),(124,152,130,146),(125,151,131,145),(126,150,132,156)], [(1,144,38,172),(2,133,39,173),(3,134,40,174),(4,135,41,175),(5,136,42,176),(6,137,43,177),(7,138,44,178),(8,139,45,179),(9,140,46,180),(10,141,47,169),(11,142,48,170),(12,143,37,171),(13,106,28,84),(14,107,29,73),(15,108,30,74),(16,97,31,75),(17,98,32,76),(18,99,33,77),(19,100,34,78),(20,101,35,79),(21,102,36,80),(22,103,25,81),(23,104,26,82),(24,105,27,83),(49,188,121,166),(50,189,122,167),(51,190,123,168),(52,191,124,157),(53,192,125,158),(54,181,126,159),(55,182,127,160),(56,183,128,161),(57,184,129,162),(58,185,130,163),(59,186,131,164),(60,187,132,165),(61,115,96,145),(62,116,85,146),(63,117,86,147),(64,118,87,148),(65,119,88,149),(66,120,89,150),(67,109,90,151),(68,110,91,152),(69,111,92,153),(70,112,93,154),(71,113,94,155),(72,114,95,156)]])
60 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3 | 4A | ··· | 4L | 4M | ··· | 4T | 6A | ··· | 6G | 12A | ··· | 12X |
| order | 1 | 2 | ··· | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | + | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | Q8 | D6 | C4○D4 | Dic6 | C4×S3 | D12 | C3⋊D4 | C4○D12 |
| kernel | (C2×Dic6)⋊7C4 | C6.C42 | C2×C4⋊Dic3 | C2×C4×C12 | C22×Dic6 | C2×Dic6 | C2×C42 | C2×C12 | C2×C12 | C22×C4 | C2×C6 | C2×C4 | C2×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 1 | 4 | 4 | 3 | 4 | 8 | 4 | 4 | 4 | 8 |
Matrix representation of (C2×Dic6)⋊7C4 ►in GL5(𝔽13)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 3 | 10 | 0 | 0 |
| 0 | 3 | 6 | 0 | 0 |
| 0 | 0 | 0 | 8 | 10 |
| 0 | 0 | 0 | 0 | 5 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 11 | 9 |
| 0 | 0 | 0 | 11 | 2 |
| 8 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 5 | 3 |
| 0 | 0 | 0 | 0 | 8 |
G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,3,3,0,0,0,10,6,0,0,0,0,0,8,0,0,0,0,10,5],[1,0,0,0,0,0,5,0,0,0,0,5,8,0,0,0,0,0,11,11,0,0,0,9,2],[8,0,0,0,0,0,5,0,0,0,0,0,5,0,0,0,0,0,5,0,0,0,0,3,8] >;
(C2×Dic6)⋊7C4 in GAP, Magma, Sage, TeX
(C_2\times {\rm Dic}_6)\rtimes_7C_4 % in TeX
G:=Group("(C2xDic6):7C4"); // GroupNames label
G:=SmallGroup(192,488);
// by ID
G=gap.SmallGroup(192,488);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,120,758,58,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=d^4=1,c^2=b^6,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=a*b^6*c>;
// generators/relations