metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.147D6, C6.272- 1+4, C4⋊C4.110D6, C42.C2.7S3, (C2×C6).230C24, C2.56(Q8○D12), Dic3.Q8.3C2, (C4×C12).223C22, (C2×C12).187C23, C3⋊(C22.58C24), C12.6Q8.12C2, C4.Dic6.13C2, Dic3⋊C4.85C22, C4⋊Dic3.237C22, C22.251(S3×C23), C2.28(Q8.15D6), (C2×Dic3).120C23, (C4×Dic3).138C22, (C3×C42.C2).6C2, (C3×C4⋊C4).185C22, (C2×C4).202(C22×S3), SmallGroup(192,1245)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.147D6
G = < a,b,c,d | a4=b4=1, c6=a2, d2=a2b2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c5 >
Subgroups: 336 in 172 conjugacy classes, 91 normal (13 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C2×C4, C2×C4, C2×C4, Dic3, C12, C2×C6, C42, C42, C4⋊C4, C4⋊C4, C2×Dic3, C2×C12, C2×C12, C42.C2, C42.C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4×C12, C3×C4⋊C4, C22.58C24, C12.6Q8, Dic3.Q8, C4.Dic6, C3×C42.C2, C42.147D6
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2- 1+4, S3×C23, C22.58C24, Q8.15D6, Q8○D12, C42.147D6
►Regular action on 192 points
Generators in S192
(1 49 7 55)(2 104 8 98)(3 51 9 57)(4 106 10 100)(5 53 11 59)(6 108 12 102)(13 130 19 124)(14 40 20 46)(15 132 21 126)(16 42 22 48)(17 122 23 128)(18 44 24 38)(25 160 31 166)(26 67 32 61)(27 162 33 168)(28 69 34 63)(29 164 35 158)(30 71 36 65)(37 112 43 118)(39 114 45 120)(41 116 47 110)(50 172 56 178)(52 174 58 180)(54 176 60 170)(62 133 68 139)(64 135 70 141)(66 137 72 143)(73 145 79 151)(74 86 80 92)(75 147 81 153)(76 88 82 94)(77 149 83 155)(78 90 84 96)(85 190 91 184)(87 192 93 186)(89 182 95 188)(97 177 103 171)(99 179 105 173)(101 169 107 175)(109 131 115 125)(111 121 117 127)(113 123 119 129)(134 163 140 157)(136 165 142 159)(138 167 144 161)(146 191 152 185)(148 181 154 187)(150 183 156 189)
(1 38 171 123)(2 124 172 39)(3 40 173 125)(4 126 174 41)(5 42 175 127)(6 128 176 43)(7 44 177 129)(8 130 178 45)(9 46 179 131)(10 132 180 47)(11 48 169 121)(12 122 170 37)(13 56 114 104)(14 105 115 57)(15 58 116 106)(16 107 117 59)(17 60 118 108)(18 97 119 49)(19 50 120 98)(20 99 109 51)(21 52 110 100)(22 101 111 53)(23 54 112 102)(24 103 113 55)(25 89 137 155)(26 156 138 90)(27 91 139 145)(28 146 140 92)(29 93 141 147)(30 148 142 94)(31 95 143 149)(32 150 144 96)(33 85 133 151)(34 152 134 86)(35 87 135 153)(36 154 136 88)(61 183 161 78)(62 79 162 184)(63 185 163 80)(64 81 164 186)(65 187 165 82)(66 83 166 188)(67 189 167 84)(68 73 168 190)(69 191 157 74)(70 75 158 192)(71 181 159 76)(72 77 160 182)
(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)
(1 165 177 71)(2 158 178 64)(3 163 179 69)(4 168 180 62)(5 161 169 67)(6 166 170 72)(7 159 171 65)(8 164 172 70)(9 157 173 63)(10 162 174 68)(11 167 175 61)(12 160 176 66)(13 147 120 87)(14 152 109 92)(15 145 110 85)(16 150 111 90)(17 155 112 95)(18 148 113 88)(19 153 114 93)(20 146 115 86)(21 151 116 91)(22 156 117 96)(23 149 118 89)(24 154 119 94)(25 60 143 102)(26 53 144 107)(27 58 133 100)(28 51 134 105)(29 56 135 98)(30 49 136 103)(31 54 137 108)(32 59 138 101)(33 52 139 106)(34 57 140 99)(35 50 141 104)(36 55 142 97)(37 188 128 77)(38 181 129 82)(39 186 130 75)(40 191 131 80)(41 184 132 73)(42 189 121 78)(43 182 122 83)(44 187 123 76)(45 192 124 81)(46 185 125 74)(47 190 126 79)(48 183 127 84)
G:=sub<Sym(192)| (1,49,7,55)(2,104,8,98)(3,51,9,57)(4,106,10,100)(5,53,11,59)(6,108,12,102)(13,130,19,124)(14,40,20,46)(15,132,21,126)(16,42,22,48)(17,122,23,128)(18,44,24,38)(25,160,31,166)(26,67,32,61)(27,162,33,168)(28,69,34,63)(29,164,35,158)(30,71,36,65)(37,112,43,118)(39,114,45,120)(41,116,47,110)(50,172,56,178)(52,174,58,180)(54,176,60,170)(62,133,68,139)(64,135,70,141)(66,137,72,143)(73,145,79,151)(74,86,80,92)(75,147,81,153)(76,88,82,94)(77,149,83,155)(78,90,84,96)(85,190,91,184)(87,192,93,186)(89,182,95,188)(97,177,103,171)(99,179,105,173)(101,169,107,175)(109,131,115,125)(111,121,117,127)(113,123,119,129)(134,163,140,157)(136,165,142,159)(138,167,144,161)(146,191,152,185)(148,181,154,187)(150,183,156,189), (1,38,171,123)(2,124,172,39)(3,40,173,125)(4,126,174,41)(5,42,175,127)(6,128,176,43)(7,44,177,129)(8,130,178,45)(9,46,179,131)(10,132,180,47)(11,48,169,121)(12,122,170,37)(13,56,114,104)(14,105,115,57)(15,58,116,106)(16,107,117,59)(17,60,118,108)(18,97,119,49)(19,50,120,98)(20,99,109,51)(21,52,110,100)(22,101,111,53)(23,54,112,102)(24,103,113,55)(25,89,137,155)(26,156,138,90)(27,91,139,145)(28,146,140,92)(29,93,141,147)(30,148,142,94)(31,95,143,149)(32,150,144,96)(33,85,133,151)(34,152,134,86)(35,87,135,153)(36,154,136,88)(61,183,161,78)(62,79,162,184)(63,185,163,80)(64,81,164,186)(65,187,165,82)(66,83,166,188)(67,189,167,84)(68,73,168,190)(69,191,157,74)(70,75,158,192)(71,181,159,76)(72,77,160,182), (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), (1,165,177,71)(2,158,178,64)(3,163,179,69)(4,168,180,62)(5,161,169,67)(6,166,170,72)(7,159,171,65)(8,164,172,70)(9,157,173,63)(10,162,174,68)(11,167,175,61)(12,160,176,66)(13,147,120,87)(14,152,109,92)(15,145,110,85)(16,150,111,90)(17,155,112,95)(18,148,113,88)(19,153,114,93)(20,146,115,86)(21,151,116,91)(22,156,117,96)(23,149,118,89)(24,154,119,94)(25,60,143,102)(26,53,144,107)(27,58,133,100)(28,51,134,105)(29,56,135,98)(30,49,136,103)(31,54,137,108)(32,59,138,101)(33,52,139,106)(34,57,140,99)(35,50,141,104)(36,55,142,97)(37,188,128,77)(38,181,129,82)(39,186,130,75)(40,191,131,80)(41,184,132,73)(42,189,121,78)(43,182,122,83)(44,187,123,76)(45,192,124,81)(46,185,125,74)(47,190,126,79)(48,183,127,84)>;
G:=Group( (1,49,7,55)(2,104,8,98)(3,51,9,57)(4,106,10,100)(5,53,11,59)(6,108,12,102)(13,130,19,124)(14,40,20,46)(15,132,21,126)(16,42,22,48)(17,122,23,128)(18,44,24,38)(25,160,31,166)(26,67,32,61)(27,162,33,168)(28,69,34,63)(29,164,35,158)(30,71,36,65)(37,112,43,118)(39,114,45,120)(41,116,47,110)(50,172,56,178)(52,174,58,180)(54,176,60,170)(62,133,68,139)(64,135,70,141)(66,137,72,143)(73,145,79,151)(74,86,80,92)(75,147,81,153)(76,88,82,94)(77,149,83,155)(78,90,84,96)(85,190,91,184)(87,192,93,186)(89,182,95,188)(97,177,103,171)(99,179,105,173)(101,169,107,175)(109,131,115,125)(111,121,117,127)(113,123,119,129)(134,163,140,157)(136,165,142,159)(138,167,144,161)(146,191,152,185)(148,181,154,187)(150,183,156,189), (1,38,171,123)(2,124,172,39)(3,40,173,125)(4,126,174,41)(5,42,175,127)(6,128,176,43)(7,44,177,129)(8,130,178,45)(9,46,179,131)(10,132,180,47)(11,48,169,121)(12,122,170,37)(13,56,114,104)(14,105,115,57)(15,58,116,106)(16,107,117,59)(17,60,118,108)(18,97,119,49)(19,50,120,98)(20,99,109,51)(21,52,110,100)(22,101,111,53)(23,54,112,102)(24,103,113,55)(25,89,137,155)(26,156,138,90)(27,91,139,145)(28,146,140,92)(29,93,141,147)(30,148,142,94)(31,95,143,149)(32,150,144,96)(33,85,133,151)(34,152,134,86)(35,87,135,153)(36,154,136,88)(61,183,161,78)(62,79,162,184)(63,185,163,80)(64,81,164,186)(65,187,165,82)(66,83,166,188)(67,189,167,84)(68,73,168,190)(69,191,157,74)(70,75,158,192)(71,181,159,76)(72,77,160,182), (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), (1,165,177,71)(2,158,178,64)(3,163,179,69)(4,168,180,62)(5,161,169,67)(6,166,170,72)(7,159,171,65)(8,164,172,70)(9,157,173,63)(10,162,174,68)(11,167,175,61)(12,160,176,66)(13,147,120,87)(14,152,109,92)(15,145,110,85)(16,150,111,90)(17,155,112,95)(18,148,113,88)(19,153,114,93)(20,146,115,86)(21,151,116,91)(22,156,117,96)(23,149,118,89)(24,154,119,94)(25,60,143,102)(26,53,144,107)(27,58,133,100)(28,51,134,105)(29,56,135,98)(30,49,136,103)(31,54,137,108)(32,59,138,101)(33,52,139,106)(34,57,140,99)(35,50,141,104)(36,55,142,97)(37,188,128,77)(38,181,129,82)(39,186,130,75)(40,191,131,80)(41,184,132,73)(42,189,121,78)(43,182,122,83)(44,187,123,76)(45,192,124,81)(46,185,125,74)(47,190,126,79)(48,183,127,84) );
G=PermutationGroup([[(1,49,7,55),(2,104,8,98),(3,51,9,57),(4,106,10,100),(5,53,11,59),(6,108,12,102),(13,130,19,124),(14,40,20,46),(15,132,21,126),(16,42,22,48),(17,122,23,128),(18,44,24,38),(25,160,31,166),(26,67,32,61),(27,162,33,168),(28,69,34,63),(29,164,35,158),(30,71,36,65),(37,112,43,118),(39,114,45,120),(41,116,47,110),(50,172,56,178),(52,174,58,180),(54,176,60,170),(62,133,68,139),(64,135,70,141),(66,137,72,143),(73,145,79,151),(74,86,80,92),(75,147,81,153),(76,88,82,94),(77,149,83,155),(78,90,84,96),(85,190,91,184),(87,192,93,186),(89,182,95,188),(97,177,103,171),(99,179,105,173),(101,169,107,175),(109,131,115,125),(111,121,117,127),(113,123,119,129),(134,163,140,157),(136,165,142,159),(138,167,144,161),(146,191,152,185),(148,181,154,187),(150,183,156,189)], [(1,38,171,123),(2,124,172,39),(3,40,173,125),(4,126,174,41),(5,42,175,127),(6,128,176,43),(7,44,177,129),(8,130,178,45),(9,46,179,131),(10,132,180,47),(11,48,169,121),(12,122,170,37),(13,56,114,104),(14,105,115,57),(15,58,116,106),(16,107,117,59),(17,60,118,108),(18,97,119,49),(19,50,120,98),(20,99,109,51),(21,52,110,100),(22,101,111,53),(23,54,112,102),(24,103,113,55),(25,89,137,155),(26,156,138,90),(27,91,139,145),(28,146,140,92),(29,93,141,147),(30,148,142,94),(31,95,143,149),(32,150,144,96),(33,85,133,151),(34,152,134,86),(35,87,135,153),(36,154,136,88),(61,183,161,78),(62,79,162,184),(63,185,163,80),(64,81,164,186),(65,187,165,82),(66,83,166,188),(67,189,167,84),(68,73,168,190),(69,191,157,74),(70,75,158,192),(71,181,159,76),(72,77,160,182)], [(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)], [(1,165,177,71),(2,158,178,64),(3,163,179,69),(4,168,180,62),(5,161,169,67),(6,166,170,72),(7,159,171,65),(8,164,172,70),(9,157,173,63),(10,162,174,68),(11,167,175,61),(12,160,176,66),(13,147,120,87),(14,152,109,92),(15,145,110,85),(16,150,111,90),(17,155,112,95),(18,148,113,88),(19,153,114,93),(20,146,115,86),(21,151,116,91),(22,156,117,96),(23,149,118,89),(24,154,119,94),(25,60,143,102),(26,53,144,107),(27,58,133,100),(28,51,134,105),(29,56,135,98),(30,49,136,103),(31,54,137,108),(32,59,138,101),(33,52,139,106),(34,57,140,99),(35,50,141,104),(36,55,142,97),(37,188,128,77),(38,181,129,82),(39,186,130,75),(40,191,131,80),(41,184,132,73),(42,189,121,78),(43,182,122,83),(44,187,123,76),(45,192,124,81),(46,185,125,74),(47,190,126,79),(48,183,127,84)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | ··· | 4G | 4H | ··· | 4O | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 4 | ··· | 4 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | 2- 1+4 | Q8.15D6 | Q8○D12 |
| kernel | C42.147D6 | C12.6Q8 | Dic3.Q8 | C4.Dic6 | C3×C42.C2 | C42.C2 | C42 | C4⋊C4 | C6 | C2 | C2 |
| # reps | 1 | 2 | 8 | 4 | 1 | 1 | 1 | 6 | 3 | 2 | 4 |
Matrix representation of C42.147D6 ►in GL8(𝔽13)
| 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 6 | 5 |
| 0 | 0 | 0 | 0 | 0 | 12 | 10 | 6 |
| 0 | 0 | 0 | 0 | 7 | 0 | 11 | 4 |
| 0 | 0 | 0 | 0 | 10 | 0 | 5 | 3 |
| 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 1 |
| 0 | 0 | 0 | 0 | 0 | 8 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 7 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 | 2 | 6 |
| 0 | 0 | 0 | 0 | 3 | 7 | 11 | 6 |
| 0 | 0 | 0 | 0 | 8 | 5 | 2 | 1 |
| 0 | 0 | 0 | 0 | 12 | 7 | 2 | 12 |
| 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 9 | 8 | 7 |
| 0 | 0 | 0 | 0 | 12 | 8 | 8 | 6 |
| 0 | 0 | 0 | 0 | 1 | 2 | 11 | 5 |
| 0 | 0 | 0 | 0 | 11 | 1 | 2 | 6 |
G:=sub<GL(8,GF(13))| [8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,7,10,0,0,0,0,0,12,0,0,0,0,0,0,6,10,11,5,0,0,0,0,5,6,4,3],[8,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,8,0,0,0,0,0,0,9,1,5,7,0,0,0,0,1,0,0,0],[0,4,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,5,3,8,12,0,0,0,0,0,7,5,7,0,0,0,0,2,11,2,2,0,0,0,0,6,6,1,12],[0,0,0,9,0,0,0,0,0,0,4,0,0,0,0,0,0,10,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,1,12,1,11,0,0,0,0,9,8,2,1,0,0,0,0,8,8,11,2,0,0,0,0,7,6,5,6] >;
C42.147D6 in GAP, Magma, Sage, TeX
C_4^2._{147}D_6 % in TeX
G:=Group("C4^2.147D6"); // GroupNames label
G:=SmallGroup(192,1245);
// by ID
G=gap.SmallGroup(192,1245);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,120,758,555,100,675,570,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=a^2,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^5>;
// generators/relations