metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C12)⋊1Q8, (C2×C4)⋊1Dic6, C6.1C22≀C2, (C2×Dic3)⋊1Q8, (C2×C4).16D12, (C2×C12).27D4, C2.6(C12⋊Q8), C6.11(C4⋊Q8), (C2×Dic3).7D4, (C22×C4).83D6, C2.6(D6⋊D4), C22.38(S3×Q8), C2.6(C4.D12), C2.4(C12⋊2Q8), C22.150(S3×D4), C22.77(C2×D12), C6.20(C22⋊Q8), (C22×Dic6).2C2, C22.40(C2×Dic6), C2.C42.13S3, C6.C42.24C2, C23.362(C22×S3), (C22×C6).281C23, (C22×C12).44C22, C22.81(D4⋊2S3), C2.8(Dic3.D4), C3⋊1(C23.78C23), (C22×Dic3).6C22, (C2×C6).93(C2×D4), (C2×C6).62(C2×Q8), (C2×C4⋊Dic3).6C2, (C2×C6).126(C4○D4), (C2×Dic3⋊C4).18C2, (C3×C2.C42).8C2, SmallGroup(192,215)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C4)⋊Dic6
G = < a,b,c,d | a2=b4=c12=1, d2=c6, cbc-1=ab=ba, ac=ca, ad=da, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 464 in 182 conjugacy classes, 69 normal (25 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C2×C4, Q8, C23, Dic3, C12, C2×C6, C2×C6, 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, Dic3⋊C4, C4⋊Dic3, C2×Dic6, C22×Dic3, C22×Dic3, C22×C12, C22×C12, C23.78C23, C6.C42, C3×C2.C42, C2×Dic3⋊C4, C2×C4⋊Dic3, C22×Dic6, (C2×C4)⋊Dic6
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, Dic6, D12, C22×S3, C22≀C2, C22⋊Q8, C4⋊Q8, C2×Dic6, C2×D12, S3×D4, D4⋊2S3, S3×Q8, C23.78C23, C12⋊2Q8, Dic3.D4, D6⋊D4, C12⋊Q8, C4.D12, (C2×C4)⋊Dic6
►Regular action on 192 points
Generators in S192
(1 62)(2 63)(3 64)(4 65)(5 66)(6 67)(7 68)(8 69)(9 70)(10 71)(11 72)(12 61)(13 124)(14 125)(15 126)(16 127)(17 128)(18 129)(19 130)(20 131)(21 132)(22 121)(23 122)(24 123)(25 95)(26 96)(27 85)(28 86)(29 87)(30 88)(31 89)(32 90)(33 91)(34 92)(35 93)(36 94)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 49)(44 50)(45 51)(46 52)(47 53)(48 54)(73 99)(74 100)(75 101)(76 102)(77 103)(78 104)(79 105)(80 106)(81 107)(82 108)(83 97)(84 98)(109 161)(110 162)(111 163)(112 164)(113 165)(114 166)(115 167)(116 168)(117 157)(118 158)(119 159)(120 160)(133 184)(134 185)(135 186)(136 187)(137 188)(138 189)(139 190)(140 191)(141 192)(142 181)(143 182)(144 183)(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 38 106 178)(2 57 107 152)(3 40 108 180)(4 59 97 154)(5 42 98 170)(6 49 99 156)(7 44 100 172)(8 51 101 146)(9 46 102 174)(10 53 103 148)(11 48 104 176)(12 55 105 150)(13 31 184 165)(14 90 185 114)(15 33 186 167)(16 92 187 116)(17 35 188 157)(18 94 189 118)(19 25 190 159)(20 96 191 120)(21 27 192 161)(22 86 181 110)(23 29 182 163)(24 88 183 112)(26 140 160 131)(28 142 162 121)(30 144 164 123)(32 134 166 125)(34 136 168 127)(36 138 158 129)(37 79 177 61)(39 81 179 63)(41 83 169 65)(43 73 171 67)(45 75 173 69)(47 77 175 71)(50 74 145 68)(52 76 147 70)(54 78 149 72)(56 80 151 62)(58 82 153 64)(60 84 155 66)(85 141 109 132)(87 143 111 122)(89 133 113 124)(91 135 115 126)(93 137 117 128)(95 139 119 130)
(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)
(1 119 7 113)(2 118 8 112)(3 117 9 111)(4 116 10 110)(5 115 11 109)(6 114 12 120)(13 151 19 145)(14 150 20 156)(15 149 21 155)(16 148 22 154)(17 147 23 153)(18 146 24 152)(25 74 31 80)(26 73 32 79)(27 84 33 78)(28 83 34 77)(29 82 35 76)(30 81 36 75)(37 140 43 134)(38 139 44 133)(39 138 45 144)(40 137 46 143)(41 136 47 142)(42 135 48 141)(49 185 55 191)(50 184 56 190)(51 183 57 189)(52 182 58 188)(53 181 59 187)(54 192 60 186)(61 160 67 166)(62 159 68 165)(63 158 69 164)(64 157 70 163)(65 168 71 162)(66 167 72 161)(85 98 91 104)(86 97 92 103)(87 108 93 102)(88 107 94 101)(89 106 95 100)(90 105 96 99)(121 169 127 175)(122 180 128 174)(123 179 129 173)(124 178 130 172)(125 177 131 171)(126 176 132 170)
G:=sub<Sym(192)| (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,71)(11,72)(12,61)(13,124)(14,125)(15,126)(16,127)(17,128)(18,129)(19,130)(20,131)(21,132)(22,121)(23,122)(24,123)(25,95)(26,96)(27,85)(28,86)(29,87)(30,88)(31,89)(32,90)(33,91)(34,92)(35,93)(36,94)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,49)(44,50)(45,51)(46,52)(47,53)(48,54)(73,99)(74,100)(75,101)(76,102)(77,103)(78,104)(79,105)(80,106)(81,107)(82,108)(83,97)(84,98)(109,161)(110,162)(111,163)(112,164)(113,165)(114,166)(115,167)(116,168)(117,157)(118,158)(119,159)(120,160)(133,184)(134,185)(135,186)(136,187)(137,188)(138,189)(139,190)(140,191)(141,192)(142,181)(143,182)(144,183)(145,172)(146,173)(147,174)(148,175)(149,176)(150,177)(151,178)(152,179)(153,180)(154,169)(155,170)(156,171), (1,38,106,178)(2,57,107,152)(3,40,108,180)(4,59,97,154)(5,42,98,170)(6,49,99,156)(7,44,100,172)(8,51,101,146)(9,46,102,174)(10,53,103,148)(11,48,104,176)(12,55,105,150)(13,31,184,165)(14,90,185,114)(15,33,186,167)(16,92,187,116)(17,35,188,157)(18,94,189,118)(19,25,190,159)(20,96,191,120)(21,27,192,161)(22,86,181,110)(23,29,182,163)(24,88,183,112)(26,140,160,131)(28,142,162,121)(30,144,164,123)(32,134,166,125)(34,136,168,127)(36,138,158,129)(37,79,177,61)(39,81,179,63)(41,83,169,65)(43,73,171,67)(45,75,173,69)(47,77,175,71)(50,74,145,68)(52,76,147,70)(54,78,149,72)(56,80,151,62)(58,82,153,64)(60,84,155,66)(85,141,109,132)(87,143,111,122)(89,133,113,124)(91,135,115,126)(93,137,117,128)(95,139,119,130), (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), (1,119,7,113)(2,118,8,112)(3,117,9,111)(4,116,10,110)(5,115,11,109)(6,114,12,120)(13,151,19,145)(14,150,20,156)(15,149,21,155)(16,148,22,154)(17,147,23,153)(18,146,24,152)(25,74,31,80)(26,73,32,79)(27,84,33,78)(28,83,34,77)(29,82,35,76)(30,81,36,75)(37,140,43,134)(38,139,44,133)(39,138,45,144)(40,137,46,143)(41,136,47,142)(42,135,48,141)(49,185,55,191)(50,184,56,190)(51,183,57,189)(52,182,58,188)(53,181,59,187)(54,192,60,186)(61,160,67,166)(62,159,68,165)(63,158,69,164)(64,157,70,163)(65,168,71,162)(66,167,72,161)(85,98,91,104)(86,97,92,103)(87,108,93,102)(88,107,94,101)(89,106,95,100)(90,105,96,99)(121,169,127,175)(122,180,128,174)(123,179,129,173)(124,178,130,172)(125,177,131,171)(126,176,132,170)>;
G:=Group( (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,71)(11,72)(12,61)(13,124)(14,125)(15,126)(16,127)(17,128)(18,129)(19,130)(20,131)(21,132)(22,121)(23,122)(24,123)(25,95)(26,96)(27,85)(28,86)(29,87)(30,88)(31,89)(32,90)(33,91)(34,92)(35,93)(36,94)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,49)(44,50)(45,51)(46,52)(47,53)(48,54)(73,99)(74,100)(75,101)(76,102)(77,103)(78,104)(79,105)(80,106)(81,107)(82,108)(83,97)(84,98)(109,161)(110,162)(111,163)(112,164)(113,165)(114,166)(115,167)(116,168)(117,157)(118,158)(119,159)(120,160)(133,184)(134,185)(135,186)(136,187)(137,188)(138,189)(139,190)(140,191)(141,192)(142,181)(143,182)(144,183)(145,172)(146,173)(147,174)(148,175)(149,176)(150,177)(151,178)(152,179)(153,180)(154,169)(155,170)(156,171), (1,38,106,178)(2,57,107,152)(3,40,108,180)(4,59,97,154)(5,42,98,170)(6,49,99,156)(7,44,100,172)(8,51,101,146)(9,46,102,174)(10,53,103,148)(11,48,104,176)(12,55,105,150)(13,31,184,165)(14,90,185,114)(15,33,186,167)(16,92,187,116)(17,35,188,157)(18,94,189,118)(19,25,190,159)(20,96,191,120)(21,27,192,161)(22,86,181,110)(23,29,182,163)(24,88,183,112)(26,140,160,131)(28,142,162,121)(30,144,164,123)(32,134,166,125)(34,136,168,127)(36,138,158,129)(37,79,177,61)(39,81,179,63)(41,83,169,65)(43,73,171,67)(45,75,173,69)(47,77,175,71)(50,74,145,68)(52,76,147,70)(54,78,149,72)(56,80,151,62)(58,82,153,64)(60,84,155,66)(85,141,109,132)(87,143,111,122)(89,133,113,124)(91,135,115,126)(93,137,117,128)(95,139,119,130), (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), (1,119,7,113)(2,118,8,112)(3,117,9,111)(4,116,10,110)(5,115,11,109)(6,114,12,120)(13,151,19,145)(14,150,20,156)(15,149,21,155)(16,148,22,154)(17,147,23,153)(18,146,24,152)(25,74,31,80)(26,73,32,79)(27,84,33,78)(28,83,34,77)(29,82,35,76)(30,81,36,75)(37,140,43,134)(38,139,44,133)(39,138,45,144)(40,137,46,143)(41,136,47,142)(42,135,48,141)(49,185,55,191)(50,184,56,190)(51,183,57,189)(52,182,58,188)(53,181,59,187)(54,192,60,186)(61,160,67,166)(62,159,68,165)(63,158,69,164)(64,157,70,163)(65,168,71,162)(66,167,72,161)(85,98,91,104)(86,97,92,103)(87,108,93,102)(88,107,94,101)(89,106,95,100)(90,105,96,99)(121,169,127,175)(122,180,128,174)(123,179,129,173)(124,178,130,172)(125,177,131,171)(126,176,132,170) );
G=PermutationGroup([[(1,62),(2,63),(3,64),(4,65),(5,66),(6,67),(7,68),(8,69),(9,70),(10,71),(11,72),(12,61),(13,124),(14,125),(15,126),(16,127),(17,128),(18,129),(19,130),(20,131),(21,132),(22,121),(23,122),(24,123),(25,95),(26,96),(27,85),(28,86),(29,87),(30,88),(31,89),(32,90),(33,91),(34,92),(35,93),(36,94),(37,55),(38,56),(39,57),(40,58),(41,59),(42,60),(43,49),(44,50),(45,51),(46,52),(47,53),(48,54),(73,99),(74,100),(75,101),(76,102),(77,103),(78,104),(79,105),(80,106),(81,107),(82,108),(83,97),(84,98),(109,161),(110,162),(111,163),(112,164),(113,165),(114,166),(115,167),(116,168),(117,157),(118,158),(119,159),(120,160),(133,184),(134,185),(135,186),(136,187),(137,188),(138,189),(139,190),(140,191),(141,192),(142,181),(143,182),(144,183),(145,172),(146,173),(147,174),(148,175),(149,176),(150,177),(151,178),(152,179),(153,180),(154,169),(155,170),(156,171)], [(1,38,106,178),(2,57,107,152),(3,40,108,180),(4,59,97,154),(5,42,98,170),(6,49,99,156),(7,44,100,172),(8,51,101,146),(9,46,102,174),(10,53,103,148),(11,48,104,176),(12,55,105,150),(13,31,184,165),(14,90,185,114),(15,33,186,167),(16,92,187,116),(17,35,188,157),(18,94,189,118),(19,25,190,159),(20,96,191,120),(21,27,192,161),(22,86,181,110),(23,29,182,163),(24,88,183,112),(26,140,160,131),(28,142,162,121),(30,144,164,123),(32,134,166,125),(34,136,168,127),(36,138,158,129),(37,79,177,61),(39,81,179,63),(41,83,169,65),(43,73,171,67),(45,75,173,69),(47,77,175,71),(50,74,145,68),(52,76,147,70),(54,78,149,72),(56,80,151,62),(58,82,153,64),(60,84,155,66),(85,141,109,132),(87,143,111,122),(89,133,113,124),(91,135,115,126),(93,137,117,128),(95,139,119,130)], [(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)], [(1,119,7,113),(2,118,8,112),(3,117,9,111),(4,116,10,110),(5,115,11,109),(6,114,12,120),(13,151,19,145),(14,150,20,156),(15,149,21,155),(16,148,22,154),(17,147,23,153),(18,146,24,152),(25,74,31,80),(26,73,32,79),(27,84,33,78),(28,83,34,77),(29,82,35,76),(30,81,36,75),(37,140,43,134),(38,139,44,133),(39,138,45,144),(40,137,46,143),(41,136,47,142),(42,135,48,141),(49,185,55,191),(50,184,56,190),(51,183,57,189),(52,182,58,188),(53,181,59,187),(54,192,60,186),(61,160,67,166),(62,159,68,165),(63,158,69,164),(64,157,70,163),(65,168,71,162),(66,167,72,161),(85,98,91,104),(86,97,92,103),(87,108,93,102),(88,107,94,101),(89,106,95,100),(90,105,96,99),(121,169,127,175),(122,180,128,174),(123,179,129,173),(124,178,130,172),(125,177,131,171),(126,176,132,170)]])
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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | - | + | - | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | D4 | Q8 | D6 | C4○D4 | Dic6 | D12 | S3×D4 | D4⋊2S3 | S3×Q8 |
| kernel | (C2×C4)⋊Dic6 | C6.C42 | C3×C2.C42 | C2×Dic3⋊C4 | C2×C4⋊Dic3 | C22×Dic6 | C2.C42 | C2×Dic3 | C2×Dic3 | C2×C12 | C2×C12 | C22×C4 | C2×C6 | C2×C4 | C2×C4 | C22 | C22 | C22 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 4 | 2 | 2 | 4 | 3 | 2 | 8 | 4 | 2 | 1 | 1 |
Matrix representation of (C2×C4)⋊Dic6 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 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 | 1 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 7 |
| 0 | 0 | 0 | 0 | 6 | 10 |
| 2 | 4 | 0 | 0 | 0 | 0 |
| 9 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 0 | 0 | 10 | 6 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 5 |
| 0 | 0 | 0 | 0 | 0 | 5 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,3,6,0,0,0,0,7,10],[2,9,0,0,0,0,4,11,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,3,10,0,0,0,0,3,6],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,8,0,0,0,0,0,5,5] >;
(C2×C4)⋊Dic6 in GAP, Magma, Sage, TeX
(C_2\times C_4)\rtimes {\rm Dic}_6 % in TeX
G:=Group("(C2xC4):Dic6"); // GroupNames label
G:=SmallGroup(192,215);
// by ID
G=gap.SmallGroup(192,215);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,120,254,387,58,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^12=1,d^2=c^6,c*b*c^-1=a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations