metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C4).17D12, (C2×C12).28D4, (C2×Dic3).3Q8, (C22×C4).28D6, C22.41(S3×Q8), C2.7(C4.D12), C6.1(C4.4D4), C22.78(C2×D12), C6.23(C22⋊Q8), C2.5(C42⋊7S3), C6.C42.7C2, (C22×C12).9C22, C6.13(C42.C2), C6.10(C42⋊2C2), C22.86(C4○D12), C2.10(Dic3.Q8), C2.C42.15S3, (C22×C6).284C23, C23.365(C22×S3), C6.6(C22.D4), C2.9(C23.8D6), C22.84(D4⋊2S3), C3⋊2(C23.83C23), C2.6(C23.21D6), (C22×Dic3).9C22, (C2×C6).94(C2×D4), (C2×C6).65(C2×Q8), (C2×C4⋊Dic3).8C2, (C2×C6).128(C4○D4), (C2×Dic3⋊C4).19C2, (C3×C2.C42).10C2, SmallGroup(192,218)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C4).17D12
G = < a,b,c,d | a2=b4=c12=1, d2=ab2, cbc-1=ab=ba, ac=ca, ad=da, dbd-1=ab-1, dcd-1=b2c-1 >
Subgroups: 336 in 134 conjugacy classes, 57 normal (25 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C2×C4, C23, Dic3, C12, C2×C6, C2×C6, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C2.C42, C2.C42, C2×C4⋊C4, Dic3⋊C4, C4⋊Dic3, C22×Dic3, C22×Dic3, C22×C12, C22×C12, C23.83C23, C6.C42, C3×C2.C42, C2×Dic3⋊C4, C2×C4⋊Dic3, (C2×C4).17D12
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, D12, C22×S3, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C2×D12, C4○D12, D4⋊2S3, S3×Q8, C23.83C23, C42⋊7S3, C23.8D6, C23.21D6, Dic3.Q8, C4.D12, (C2×C4).17D12
►Regular action on 192 points
Generators in S192
(1 134)(2 135)(3 136)(4 137)(5 138)(6 139)(7 140)(8 141)(9 142)(10 143)(11 144)(12 133)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 49)(22 50)(23 51)(24 52)(25 131)(26 132)(27 121)(28 122)(29 123)(30 124)(31 125)(32 126)(33 127)(34 128)(35 129)(36 130)(37 80)(38 81)(39 82)(40 83)(41 84)(42 73)(43 74)(44 75)(45 76)(46 77)(47 78)(48 79)(61 171)(62 172)(63 173)(64 174)(65 175)(66 176)(67 177)(68 178)(69 179)(70 180)(71 169)(72 170)(85 118)(86 119)(87 120)(88 109)(89 110)(90 111)(91 112)(92 113)(93 114)(94 115)(95 116)(96 117)(97 164)(98 165)(99 166)(100 167)(101 168)(102 157)(103 158)(104 159)(105 160)(106 161)(107 162)(108 163)(145 188)(146 189)(147 190)(148 191)(149 192)(150 181)(151 182)(152 183)(153 184)(154 185)(155 186)(156 187)
(1 41 100 109)(2 73 101 89)(3 43 102 111)(4 75 103 91)(5 45 104 113)(6 77 105 93)(7 47 106 115)(8 79 107 95)(9 37 108 117)(10 81 97 85)(11 39 98 119)(12 83 99 87)(13 131 172 153)(14 26 173 185)(15 121 174 155)(16 28 175 187)(17 123 176 145)(18 30 177 189)(19 125 178 147)(20 32 179 191)(21 127 180 149)(22 34 169 181)(23 129 170 151)(24 36 171 183)(25 62 184 53)(27 64 186 55)(29 66 188 57)(31 68 190 59)(33 70 192 49)(35 72 182 51)(38 164 118 143)(40 166 120 133)(42 168 110 135)(44 158 112 137)(46 160 114 139)(48 162 116 141)(50 128 71 150)(52 130 61 152)(54 132 63 154)(56 122 65 156)(58 124 67 146)(60 126 69 148)(74 157 90 136)(76 159 92 138)(78 161 94 140)(80 163 96 142)(82 165 86 144)(84 167 88 134)
(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)
(1 178 167 59)(2 18 168 67)(3 176 157 57)(4 16 158 65)(5 174 159 55)(6 14 160 63)(7 172 161 53)(8 24 162 61)(9 170 163 51)(10 22 164 71)(11 180 165 49)(12 20 166 69)(13 140 62 106)(15 138 64 104)(17 136 66 102)(19 134 68 100)(21 144 70 98)(23 142 72 108)(25 94 153 47)(26 77 154 114)(27 92 155 45)(28 75 156 112)(29 90 145 43)(30 73 146 110)(31 88 147 41)(32 83 148 120)(33 86 149 39)(34 81 150 118)(35 96 151 37)(36 79 152 116)(38 181 85 128)(40 191 87 126)(42 189 89 124)(44 187 91 122)(46 185 93 132)(48 183 95 130)(50 97 169 143)(52 107 171 141)(54 105 173 139)(56 103 175 137)(58 101 177 135)(60 99 179 133)(74 123 111 188)(76 121 113 186)(78 131 115 184)(80 129 117 182)(82 127 119 192)(84 125 109 190)
G:=sub<Sym(192)| (1,134)(2,135)(3,136)(4,137)(5,138)(6,139)(7,140)(8,141)(9,142)(10,143)(11,144)(12,133)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,49)(22,50)(23,51)(24,52)(25,131)(26,132)(27,121)(28,122)(29,123)(30,124)(31,125)(32,126)(33,127)(34,128)(35,129)(36,130)(37,80)(38,81)(39,82)(40,83)(41,84)(42,73)(43,74)(44,75)(45,76)(46,77)(47,78)(48,79)(61,171)(62,172)(63,173)(64,174)(65,175)(66,176)(67,177)(68,178)(69,179)(70,180)(71,169)(72,170)(85,118)(86,119)(87,120)(88,109)(89,110)(90,111)(91,112)(92,113)(93,114)(94,115)(95,116)(96,117)(97,164)(98,165)(99,166)(100,167)(101,168)(102,157)(103,158)(104,159)(105,160)(106,161)(107,162)(108,163)(145,188)(146,189)(147,190)(148,191)(149,192)(150,181)(151,182)(152,183)(153,184)(154,185)(155,186)(156,187), (1,41,100,109)(2,73,101,89)(3,43,102,111)(4,75,103,91)(5,45,104,113)(6,77,105,93)(7,47,106,115)(8,79,107,95)(9,37,108,117)(10,81,97,85)(11,39,98,119)(12,83,99,87)(13,131,172,153)(14,26,173,185)(15,121,174,155)(16,28,175,187)(17,123,176,145)(18,30,177,189)(19,125,178,147)(20,32,179,191)(21,127,180,149)(22,34,169,181)(23,129,170,151)(24,36,171,183)(25,62,184,53)(27,64,186,55)(29,66,188,57)(31,68,190,59)(33,70,192,49)(35,72,182,51)(38,164,118,143)(40,166,120,133)(42,168,110,135)(44,158,112,137)(46,160,114,139)(48,162,116,141)(50,128,71,150)(52,130,61,152)(54,132,63,154)(56,122,65,156)(58,124,67,146)(60,126,69,148)(74,157,90,136)(76,159,92,138)(78,161,94,140)(80,163,96,142)(82,165,86,144)(84,167,88,134), (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), (1,178,167,59)(2,18,168,67)(3,176,157,57)(4,16,158,65)(5,174,159,55)(6,14,160,63)(7,172,161,53)(8,24,162,61)(9,170,163,51)(10,22,164,71)(11,180,165,49)(12,20,166,69)(13,140,62,106)(15,138,64,104)(17,136,66,102)(19,134,68,100)(21,144,70,98)(23,142,72,108)(25,94,153,47)(26,77,154,114)(27,92,155,45)(28,75,156,112)(29,90,145,43)(30,73,146,110)(31,88,147,41)(32,83,148,120)(33,86,149,39)(34,81,150,118)(35,96,151,37)(36,79,152,116)(38,181,85,128)(40,191,87,126)(42,189,89,124)(44,187,91,122)(46,185,93,132)(48,183,95,130)(50,97,169,143)(52,107,171,141)(54,105,173,139)(56,103,175,137)(58,101,177,135)(60,99,179,133)(74,123,111,188)(76,121,113,186)(78,131,115,184)(80,129,117,182)(82,127,119,192)(84,125,109,190)>;
G:=Group( (1,134)(2,135)(3,136)(4,137)(5,138)(6,139)(7,140)(8,141)(9,142)(10,143)(11,144)(12,133)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,49)(22,50)(23,51)(24,52)(25,131)(26,132)(27,121)(28,122)(29,123)(30,124)(31,125)(32,126)(33,127)(34,128)(35,129)(36,130)(37,80)(38,81)(39,82)(40,83)(41,84)(42,73)(43,74)(44,75)(45,76)(46,77)(47,78)(48,79)(61,171)(62,172)(63,173)(64,174)(65,175)(66,176)(67,177)(68,178)(69,179)(70,180)(71,169)(72,170)(85,118)(86,119)(87,120)(88,109)(89,110)(90,111)(91,112)(92,113)(93,114)(94,115)(95,116)(96,117)(97,164)(98,165)(99,166)(100,167)(101,168)(102,157)(103,158)(104,159)(105,160)(106,161)(107,162)(108,163)(145,188)(146,189)(147,190)(148,191)(149,192)(150,181)(151,182)(152,183)(153,184)(154,185)(155,186)(156,187), (1,41,100,109)(2,73,101,89)(3,43,102,111)(4,75,103,91)(5,45,104,113)(6,77,105,93)(7,47,106,115)(8,79,107,95)(9,37,108,117)(10,81,97,85)(11,39,98,119)(12,83,99,87)(13,131,172,153)(14,26,173,185)(15,121,174,155)(16,28,175,187)(17,123,176,145)(18,30,177,189)(19,125,178,147)(20,32,179,191)(21,127,180,149)(22,34,169,181)(23,129,170,151)(24,36,171,183)(25,62,184,53)(27,64,186,55)(29,66,188,57)(31,68,190,59)(33,70,192,49)(35,72,182,51)(38,164,118,143)(40,166,120,133)(42,168,110,135)(44,158,112,137)(46,160,114,139)(48,162,116,141)(50,128,71,150)(52,130,61,152)(54,132,63,154)(56,122,65,156)(58,124,67,146)(60,126,69,148)(74,157,90,136)(76,159,92,138)(78,161,94,140)(80,163,96,142)(82,165,86,144)(84,167,88,134), (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), (1,178,167,59)(2,18,168,67)(3,176,157,57)(4,16,158,65)(5,174,159,55)(6,14,160,63)(7,172,161,53)(8,24,162,61)(9,170,163,51)(10,22,164,71)(11,180,165,49)(12,20,166,69)(13,140,62,106)(15,138,64,104)(17,136,66,102)(19,134,68,100)(21,144,70,98)(23,142,72,108)(25,94,153,47)(26,77,154,114)(27,92,155,45)(28,75,156,112)(29,90,145,43)(30,73,146,110)(31,88,147,41)(32,83,148,120)(33,86,149,39)(34,81,150,118)(35,96,151,37)(36,79,152,116)(38,181,85,128)(40,191,87,126)(42,189,89,124)(44,187,91,122)(46,185,93,132)(48,183,95,130)(50,97,169,143)(52,107,171,141)(54,105,173,139)(56,103,175,137)(58,101,177,135)(60,99,179,133)(74,123,111,188)(76,121,113,186)(78,131,115,184)(80,129,117,182)(82,127,119,192)(84,125,109,190) );
G=PermutationGroup([[(1,134),(2,135),(3,136),(4,137),(5,138),(6,139),(7,140),(8,141),(9,142),(10,143),(11,144),(12,133),(13,53),(14,54),(15,55),(16,56),(17,57),(18,58),(19,59),(20,60),(21,49),(22,50),(23,51),(24,52),(25,131),(26,132),(27,121),(28,122),(29,123),(30,124),(31,125),(32,126),(33,127),(34,128),(35,129),(36,130),(37,80),(38,81),(39,82),(40,83),(41,84),(42,73),(43,74),(44,75),(45,76),(46,77),(47,78),(48,79),(61,171),(62,172),(63,173),(64,174),(65,175),(66,176),(67,177),(68,178),(69,179),(70,180),(71,169),(72,170),(85,118),(86,119),(87,120),(88,109),(89,110),(90,111),(91,112),(92,113),(93,114),(94,115),(95,116),(96,117),(97,164),(98,165),(99,166),(100,167),(101,168),(102,157),(103,158),(104,159),(105,160),(106,161),(107,162),(108,163),(145,188),(146,189),(147,190),(148,191),(149,192),(150,181),(151,182),(152,183),(153,184),(154,185),(155,186),(156,187)], [(1,41,100,109),(2,73,101,89),(3,43,102,111),(4,75,103,91),(5,45,104,113),(6,77,105,93),(7,47,106,115),(8,79,107,95),(9,37,108,117),(10,81,97,85),(11,39,98,119),(12,83,99,87),(13,131,172,153),(14,26,173,185),(15,121,174,155),(16,28,175,187),(17,123,176,145),(18,30,177,189),(19,125,178,147),(20,32,179,191),(21,127,180,149),(22,34,169,181),(23,129,170,151),(24,36,171,183),(25,62,184,53),(27,64,186,55),(29,66,188,57),(31,68,190,59),(33,70,192,49),(35,72,182,51),(38,164,118,143),(40,166,120,133),(42,168,110,135),(44,158,112,137),(46,160,114,139),(48,162,116,141),(50,128,71,150),(52,130,61,152),(54,132,63,154),(56,122,65,156),(58,124,67,146),(60,126,69,148),(74,157,90,136),(76,159,92,138),(78,161,94,140),(80,163,96,142),(82,165,86,144),(84,167,88,134)], [(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)], [(1,178,167,59),(2,18,168,67),(3,176,157,57),(4,16,158,65),(5,174,159,55),(6,14,160,63),(7,172,161,53),(8,24,162,61),(9,170,163,51),(10,22,164,71),(11,180,165,49),(12,20,166,69),(13,140,62,106),(15,138,64,104),(17,136,66,102),(19,134,68,100),(21,144,70,98),(23,142,72,108),(25,94,153,47),(26,77,154,114),(27,92,155,45),(28,75,156,112),(29,90,145,43),(30,73,146,110),(31,88,147,41),(32,83,148,120),(33,86,149,39),(34,81,150,118),(35,96,151,37),(36,79,152,116),(38,181,85,128),(40,191,87,126),(42,189,89,124),(44,187,91,122),(46,185,93,132),(48,183,95,130),(50,97,169,143),(52,107,171,141),(54,105,173,139),(56,103,175,137),(58,101,177,135),(60,99,179,133),(74,123,111,188),(76,121,113,186),(78,131,115,184),(80,129,117,182),(82,127,119,192),(84,125,109,190)]])
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 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | S3 | Q8 | D4 | D6 | C4○D4 | D12 | C4○D12 | D4⋊2S3 | S3×Q8 |
| kernel | (C2×C4).17D12 | C6.C42 | C3×C2.C42 | C2×Dic3⋊C4 | C2×C4⋊Dic3 | C2.C42 | C2×Dic3 | C2×C12 | C22×C4 | C2×C6 | C2×C4 | C22 | C22 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 10 | 4 | 8 | 3 | 1 |
Matrix representation of (C2×C4).17D12 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 0 | 0 | 0 | 0 | 7 | 0 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[8,0,0,0,0,0,0,5,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,3,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,7,0,0,0,0,0,0,2],[0,4,0,0,0,0,3,0,0,0,0,0,0,0,0,5,0,0,0,0,8,0,0,0,0,0,0,0,0,7,0,0,0,0,2,0] >;
(C2×C4).17D12 in GAP, Magma, Sage, TeX
(C_2\times C_4)._{17}D_{12} % in TeX
G:=Group("(C2xC4).17D12"); // GroupNames label
G:=SmallGroup(192,218);
// by ID
G=gap.SmallGroup(192,218);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,64,254,387,268,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^12=1,d^2=a*b^2,c*b*c^-1=a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=a*b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations