metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C12).53D4, (C2×C4).44D12, C6.26(C4⋊Q8), (C2×Dic3).5Q8, C22.46(S3×Q8), C6.61(C4⋊D4), C2.9(C12⋊7D4), (C22×C4).115D6, C6.46(C22⋊Q8), C2.16(C4.D12), C22.128(C2×D12), C6.20(C42.C2), C2.6(Dic3⋊Q8), C2.11(Dic3.Q8), C23.32D6.28C2, C23.385(C22×S3), (C22×C12).64C22, (C22×C6).345C23, C22.103(C4○D12), C22.99(D4⋊2S3), C3⋊3(C23.81C23), C2.9(C23.23D6), C6.75(C22.D4), (C22×Dic3).53C22, (C6×C4⋊C4).22C2, (C2×C4⋊C4).18S3, (C2×C6).80(C2×Q8), (C2×C6).151(C2×D4), (C2×C6).84(C4○D4), (C2×C4).36(C3⋊D4), (C2×C4⋊Dic3).18C2, (C2×Dic3⋊C4).31C2, C22.135(C2×C3⋊D4), SmallGroup(192,540)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C12).53D4
G = < a,b,c,d | a2=b12=c4=1, d2=a, ab=ba, ac=ca, ad=da, cbc-1=ab-1, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 360 in 150 conjugacy classes, 63 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×11], C22 [×3], C22 [×4], C6 [×3], C6 [×4], C2×C4 [×4], C2×C4 [×21], C23, Dic3 [×6], C12 [×5], C2×C6 [×3], C2×C6 [×4], C4⋊C4 [×8], C22×C4 [×3], C22×C4 [×4], C2×Dic3 [×4], C2×Dic3 [×10], C2×C12 [×4], C2×C12 [×7], C22×C6, C2.C42 [×3], C2×C4⋊C4, C2×C4⋊C4 [×3], Dic3⋊C4 [×4], C4⋊Dic3 [×2], C3×C4⋊C4 [×2], C22×Dic3 [×4], C22×C12 [×3], C23.81C23, C23.32D6, C23.32D6 [×2], C2×Dic3⋊C4 [×2], C2×C4⋊Dic3, C6×C4⋊C4, (C2×C12).53D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], Q8 [×4], C23, D6 [×3], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×3], D12 [×2], C3⋊D4 [×2], C22×S3, C4⋊D4, C22⋊Q8 [×2], C22.D4, C42.C2 [×2], C4⋊Q8, C2×D12, C4○D12, D4⋊2S3 [×2], S3×Q8 [×2], C2×C3⋊D4, C23.81C23, Dic3.Q8 [×2], C4.D12 [×2], C12⋊7D4, C23.23D6, Dic3⋊Q8, (C2×C12).53D4
►Regular action on 192 points
Generators in S192
(1 192)(2 181)(3 182)(4 183)(5 184)(6 185)(7 186)(8 187)(9 188)(10 189)(11 190)(12 191)(13 44)(14 45)(15 46)(16 47)(17 48)(18 37)(19 38)(20 39)(21 40)(22 41)(23 42)(24 43)(25 95)(26 96)(27 85)(28 86)(29 87)(30 88)(31 89)(32 90)(33 91)(34 92)(35 93)(36 94)(49 101)(50 102)(51 103)(52 104)(53 105)(54 106)(55 107)(56 108)(57 97)(58 98)(59 99)(60 100)(61 154)(62 155)(63 156)(64 145)(65 146)(66 147)(67 148)(68 149)(69 150)(70 151)(71 152)(72 153)(73 159)(74 160)(75 161)(76 162)(77 163)(78 164)(79 165)(80 166)(81 167)(82 168)(83 157)(84 158)(109 179)(110 180)(111 169)(112 170)(113 171)(114 172)(115 173)(116 174)(117 175)(118 176)(119 177)(120 178)(121 142)(122 143)(123 144)(124 133)(125 134)(126 135)(127 136)(128 137)(129 138)(130 139)(131 140)(132 141)
(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)
(1 155 90 161)(2 61 91 74)(3 153 92 159)(4 71 93 84)(5 151 94 157)(6 69 95 82)(7 149 96 167)(8 67 85 80)(9 147 86 165)(10 65 87 78)(11 145 88 163)(12 63 89 76)(13 172 123 99)(14 113 124 58)(15 170 125 97)(16 111 126 56)(17 180 127 107)(18 109 128 54)(19 178 129 105)(20 119 130 52)(21 176 131 103)(22 117 132 50)(23 174 121 101)(24 115 122 60)(25 168 185 150)(26 81 186 68)(27 166 187 148)(28 79 188 66)(29 164 189 146)(30 77 190 64)(31 162 191 156)(32 75 192 62)(33 160 181 154)(34 73 182 72)(35 158 183 152)(36 83 184 70)(37 179 137 106)(38 120 138 53)(39 177 139 104)(40 118 140 51)(41 175 141 102)(42 116 142 49)(43 173 143 100)(44 114 144 59)(45 171 133 98)(46 112 134 57)(47 169 135 108)(48 110 136 55)
(1 171 192 113)(2 170 181 112)(3 169 182 111)(4 180 183 110)(5 179 184 109)(6 178 185 120)(7 177 186 119)(8 176 187 118)(9 175 188 117)(10 174 189 116)(11 173 190 115)(12 172 191 114)(13 156 44 63)(14 155 45 62)(15 154 46 61)(16 153 47 72)(17 152 48 71)(18 151 37 70)(19 150 38 69)(20 149 39 68)(21 148 40 67)(22 147 41 66)(23 146 42 65)(24 145 43 64)(25 53 95 105)(26 52 96 104)(27 51 85 103)(28 50 86 102)(29 49 87 101)(30 60 88 100)(31 59 89 99)(32 58 90 98)(33 57 91 97)(34 56 92 108)(35 55 93 107)(36 54 94 106)(73 126 159 135)(74 125 160 134)(75 124 161 133)(76 123 162 144)(77 122 163 143)(78 121 164 142)(79 132 165 141)(80 131 166 140)(81 130 167 139)(82 129 168 138)(83 128 157 137)(84 127 158 136)
G:=sub<Sym(192)| (1,192)(2,181)(3,182)(4,183)(5,184)(6,185)(7,186)(8,187)(9,188)(10,189)(11,190)(12,191)(13,44)(14,45)(15,46)(16,47)(17,48)(18,37)(19,38)(20,39)(21,40)(22,41)(23,42)(24,43)(25,95)(26,96)(27,85)(28,86)(29,87)(30,88)(31,89)(32,90)(33,91)(34,92)(35,93)(36,94)(49,101)(50,102)(51,103)(52,104)(53,105)(54,106)(55,107)(56,108)(57,97)(58,98)(59,99)(60,100)(61,154)(62,155)(63,156)(64,145)(65,146)(66,147)(67,148)(68,149)(69,150)(70,151)(71,152)(72,153)(73,159)(74,160)(75,161)(76,162)(77,163)(78,164)(79,165)(80,166)(81,167)(82,168)(83,157)(84,158)(109,179)(110,180)(111,169)(112,170)(113,171)(114,172)(115,173)(116,174)(117,175)(118,176)(119,177)(120,178)(121,142)(122,143)(123,144)(124,133)(125,134)(126,135)(127,136)(128,137)(129,138)(130,139)(131,140)(132,141), (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), (1,155,90,161)(2,61,91,74)(3,153,92,159)(4,71,93,84)(5,151,94,157)(6,69,95,82)(7,149,96,167)(8,67,85,80)(9,147,86,165)(10,65,87,78)(11,145,88,163)(12,63,89,76)(13,172,123,99)(14,113,124,58)(15,170,125,97)(16,111,126,56)(17,180,127,107)(18,109,128,54)(19,178,129,105)(20,119,130,52)(21,176,131,103)(22,117,132,50)(23,174,121,101)(24,115,122,60)(25,168,185,150)(26,81,186,68)(27,166,187,148)(28,79,188,66)(29,164,189,146)(30,77,190,64)(31,162,191,156)(32,75,192,62)(33,160,181,154)(34,73,182,72)(35,158,183,152)(36,83,184,70)(37,179,137,106)(38,120,138,53)(39,177,139,104)(40,118,140,51)(41,175,141,102)(42,116,142,49)(43,173,143,100)(44,114,144,59)(45,171,133,98)(46,112,134,57)(47,169,135,108)(48,110,136,55), (1,171,192,113)(2,170,181,112)(3,169,182,111)(4,180,183,110)(5,179,184,109)(6,178,185,120)(7,177,186,119)(8,176,187,118)(9,175,188,117)(10,174,189,116)(11,173,190,115)(12,172,191,114)(13,156,44,63)(14,155,45,62)(15,154,46,61)(16,153,47,72)(17,152,48,71)(18,151,37,70)(19,150,38,69)(20,149,39,68)(21,148,40,67)(22,147,41,66)(23,146,42,65)(24,145,43,64)(25,53,95,105)(26,52,96,104)(27,51,85,103)(28,50,86,102)(29,49,87,101)(30,60,88,100)(31,59,89,99)(32,58,90,98)(33,57,91,97)(34,56,92,108)(35,55,93,107)(36,54,94,106)(73,126,159,135)(74,125,160,134)(75,124,161,133)(76,123,162,144)(77,122,163,143)(78,121,164,142)(79,132,165,141)(80,131,166,140)(81,130,167,139)(82,129,168,138)(83,128,157,137)(84,127,158,136)>;
G:=Group( (1,192)(2,181)(3,182)(4,183)(5,184)(6,185)(7,186)(8,187)(9,188)(10,189)(11,190)(12,191)(13,44)(14,45)(15,46)(16,47)(17,48)(18,37)(19,38)(20,39)(21,40)(22,41)(23,42)(24,43)(25,95)(26,96)(27,85)(28,86)(29,87)(30,88)(31,89)(32,90)(33,91)(34,92)(35,93)(36,94)(49,101)(50,102)(51,103)(52,104)(53,105)(54,106)(55,107)(56,108)(57,97)(58,98)(59,99)(60,100)(61,154)(62,155)(63,156)(64,145)(65,146)(66,147)(67,148)(68,149)(69,150)(70,151)(71,152)(72,153)(73,159)(74,160)(75,161)(76,162)(77,163)(78,164)(79,165)(80,166)(81,167)(82,168)(83,157)(84,158)(109,179)(110,180)(111,169)(112,170)(113,171)(114,172)(115,173)(116,174)(117,175)(118,176)(119,177)(120,178)(121,142)(122,143)(123,144)(124,133)(125,134)(126,135)(127,136)(128,137)(129,138)(130,139)(131,140)(132,141), (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), (1,155,90,161)(2,61,91,74)(3,153,92,159)(4,71,93,84)(5,151,94,157)(6,69,95,82)(7,149,96,167)(8,67,85,80)(9,147,86,165)(10,65,87,78)(11,145,88,163)(12,63,89,76)(13,172,123,99)(14,113,124,58)(15,170,125,97)(16,111,126,56)(17,180,127,107)(18,109,128,54)(19,178,129,105)(20,119,130,52)(21,176,131,103)(22,117,132,50)(23,174,121,101)(24,115,122,60)(25,168,185,150)(26,81,186,68)(27,166,187,148)(28,79,188,66)(29,164,189,146)(30,77,190,64)(31,162,191,156)(32,75,192,62)(33,160,181,154)(34,73,182,72)(35,158,183,152)(36,83,184,70)(37,179,137,106)(38,120,138,53)(39,177,139,104)(40,118,140,51)(41,175,141,102)(42,116,142,49)(43,173,143,100)(44,114,144,59)(45,171,133,98)(46,112,134,57)(47,169,135,108)(48,110,136,55), (1,171,192,113)(2,170,181,112)(3,169,182,111)(4,180,183,110)(5,179,184,109)(6,178,185,120)(7,177,186,119)(8,176,187,118)(9,175,188,117)(10,174,189,116)(11,173,190,115)(12,172,191,114)(13,156,44,63)(14,155,45,62)(15,154,46,61)(16,153,47,72)(17,152,48,71)(18,151,37,70)(19,150,38,69)(20,149,39,68)(21,148,40,67)(22,147,41,66)(23,146,42,65)(24,145,43,64)(25,53,95,105)(26,52,96,104)(27,51,85,103)(28,50,86,102)(29,49,87,101)(30,60,88,100)(31,59,89,99)(32,58,90,98)(33,57,91,97)(34,56,92,108)(35,55,93,107)(36,54,94,106)(73,126,159,135)(74,125,160,134)(75,124,161,133)(76,123,162,144)(77,122,163,143)(78,121,164,142)(79,132,165,141)(80,131,166,140)(81,130,167,139)(82,129,168,138)(83,128,157,137)(84,127,158,136) );
G=PermutationGroup([(1,192),(2,181),(3,182),(4,183),(5,184),(6,185),(7,186),(8,187),(9,188),(10,189),(11,190),(12,191),(13,44),(14,45),(15,46),(16,47),(17,48),(18,37),(19,38),(20,39),(21,40),(22,41),(23,42),(24,43),(25,95),(26,96),(27,85),(28,86),(29,87),(30,88),(31,89),(32,90),(33,91),(34,92),(35,93),(36,94),(49,101),(50,102),(51,103),(52,104),(53,105),(54,106),(55,107),(56,108),(57,97),(58,98),(59,99),(60,100),(61,154),(62,155),(63,156),(64,145),(65,146),(66,147),(67,148),(68,149),(69,150),(70,151),(71,152),(72,153),(73,159),(74,160),(75,161),(76,162),(77,163),(78,164),(79,165),(80,166),(81,167),(82,168),(83,157),(84,158),(109,179),(110,180),(111,169),(112,170),(113,171),(114,172),(115,173),(116,174),(117,175),(118,176),(119,177),(120,178),(121,142),(122,143),(123,144),(124,133),(125,134),(126,135),(127,136),(128,137),(129,138),(130,139),(131,140),(132,141)], [(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)], [(1,155,90,161),(2,61,91,74),(3,153,92,159),(4,71,93,84),(5,151,94,157),(6,69,95,82),(7,149,96,167),(8,67,85,80),(9,147,86,165),(10,65,87,78),(11,145,88,163),(12,63,89,76),(13,172,123,99),(14,113,124,58),(15,170,125,97),(16,111,126,56),(17,180,127,107),(18,109,128,54),(19,178,129,105),(20,119,130,52),(21,176,131,103),(22,117,132,50),(23,174,121,101),(24,115,122,60),(25,168,185,150),(26,81,186,68),(27,166,187,148),(28,79,188,66),(29,164,189,146),(30,77,190,64),(31,162,191,156),(32,75,192,62),(33,160,181,154),(34,73,182,72),(35,158,183,152),(36,83,184,70),(37,179,137,106),(38,120,138,53),(39,177,139,104),(40,118,140,51),(41,175,141,102),(42,116,142,49),(43,173,143,100),(44,114,144,59),(45,171,133,98),(46,112,134,57),(47,169,135,108),(48,110,136,55)], [(1,171,192,113),(2,170,181,112),(3,169,182,111),(4,180,183,110),(5,179,184,109),(6,178,185,120),(7,177,186,119),(8,176,187,118),(9,175,188,117),(10,174,189,116),(11,173,190,115),(12,172,191,114),(13,156,44,63),(14,155,45,62),(15,154,46,61),(16,153,47,72),(17,152,48,71),(18,151,37,70),(19,150,38,69),(20,149,39,68),(21,148,40,67),(22,147,41,66),(23,146,42,65),(24,145,43,64),(25,53,95,105),(26,52,96,104),(27,51,85,103),(28,50,86,102),(29,49,87,101),(30,60,88,100),(31,59,89,99),(32,58,90,98),(33,57,91,97),(34,56,92,108),(35,55,93,107),(36,54,94,106),(73,126,159,135),(74,125,160,134),(75,124,161,133),(76,123,162,144),(77,122,163,143),(78,121,164,142),(79,132,165,141),(80,131,166,140),(81,130,167,139),(82,129,168,138),(83,128,157,137),(84,127,158,136)])
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 |
| type | + | + | + | + | + | + | - | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | S3 | Q8 | D4 | D6 | C4○D4 | D12 | C3⋊D4 | C4○D12 | D4⋊2S3 | S3×Q8 |
| kernel | (C2×C12).53D4 | C23.32D6 | C2×Dic3⋊C4 | C2×C4⋊Dic3 | C6×C4⋊C4 | C2×C4⋊C4 | C2×Dic3 | C2×C12 | C22×C4 | C2×C6 | C2×C4 | C2×C4 | C22 | C22 | C22 |
| # reps | 1 | 3 | 2 | 1 | 1 | 1 | 4 | 4 | 3 | 6 | 4 | 4 | 4 | 2 | 2 |
Matrix representation of (C2×C12).53D4 ►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 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 0 | 0 | 10 | 6 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 8 | 5 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 7 |
| 0 | 0 | 0 | 0 | 10 | 10 |
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],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,10,0,0,0,0,3,6],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,8,8,0,0,0,0,0,5],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,3,10,0,0,0,0,7,10] >;
(C2×C12).53D4 in GAP, Magma, Sage, TeX
(C_2\times C_{12})._{53}D_4 % in TeX
G:=Group("(C2xC12).53D4"); // GroupNames label
G:=SmallGroup(192,540);
// by ID
G=gap.SmallGroup(192,540);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,120,254,387,184,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=c^4=1,d^2=a,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^-1,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations