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, (C2×C12).187C23, (C4×C12).223C22, C3⋊(C22.58C24), C12.3Q8.13C2, C12.6Q8.12C2, Dic3⋊C4.85C22, C4⋊Dic3.237C22, C22.251(S3×C23), C2.28(Q8.15D6), (C4×Dic3).138C22, (C2×Dic3).120C23, (C3×C42.C2).6C2, (C3×C4⋊C4).185C22, (C2×C4).202(C22×S3), SmallGroup(192,1245)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 336 in 172 conjugacy classes, 91 normal (13 characteristic)
C1, C2, C2 [×2], C3, C4 [×15], C22, C6, C6 [×2], C2×C4, C2×C4 [×6], C2×C4 [×8], Dic3 [×8], C12 [×7], C2×C6, C42, C42 [×4], C4⋊C4 [×6], C4⋊C4 [×24], C2×Dic3 [×8], C2×C12, C2×C12 [×6], C42.C2, C42.C2 [×14], C4×Dic3 [×4], Dic3⋊C4 [×16], C4⋊Dic3 [×8], C4×C12, C3×C4⋊C4 [×6], C22.58C24, C12.6Q8 [×2], Dic3.Q8 [×8], C12.3Q8 [×4], C3×C42.C2, C42.147D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C24, C22×S3 [×7], 2- (1+4) [×3], S3×C23, C22.58C24, Q8.15D6, Q8○D12 [×2], C42.147D6
Generators and relations
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 >
►Regular action on 192 points
Generators in S192
(1 159 7 165)(2 42 8 48)(3 161 9 167)(4 44 10 38)(5 163 11 157)(6 46 12 40)(13 43 19 37)(14 162 20 168)(15 45 21 39)(16 164 22 158)(17 47 23 41)(18 166 24 160)(25 112 31 118)(26 100 32 106)(27 114 33 120)(28 102 34 108)(29 116 35 110)(30 104 36 98)(49 146 55 152)(50 123 56 129)(51 148 57 154)(52 125 58 131)(53 150 59 156)(54 127 60 121)(61 109 67 115)(62 97 68 103)(63 111 69 117)(64 99 70 105)(65 113 71 119)(66 101 72 107)(73 145 79 151)(74 122 80 128)(75 147 81 153)(76 124 82 130)(77 149 83 155)(78 126 84 132)(85 188 91 182)(86 174 92 180)(87 190 93 184)(88 176 94 170)(89 192 95 186)(90 178 96 172)(133 169 139 175)(134 185 140 191)(135 171 141 177)(136 187 142 181)(137 173 143 179)(138 189 144 183)
(1 105 17 112)(2 113 18 106)(3 107 19 114)(4 115 20 108)(5 97 21 116)(6 117 22 98)(7 99 23 118)(8 119 24 100)(9 101 13 120)(10 109 14 102)(11 103 15 110)(12 111 16 104)(25 165 70 41)(26 42 71 166)(27 167 72 43)(28 44 61 168)(29 157 62 45)(30 46 63 158)(31 159 64 47)(32 48 65 160)(33 161 66 37)(34 38 67 162)(35 163 68 39)(36 40 69 164)(49 176 80 185)(50 186 81 177)(51 178 82 187)(52 188 83 179)(53 180 84 189)(54 190 73 169)(55 170 74 191)(56 192 75 171)(57 172 76 181)(58 182 77 173)(59 174 78 183)(60 184 79 175)(85 149 143 131)(86 132 144 150)(87 151 133 121)(88 122 134 152)(89 153 135 123)(90 124 136 154)(91 155 137 125)(92 126 138 156)(93 145 139 127)(94 128 140 146)(95 147 141 129)(96 130 142 148)
(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)
(1 140 23 88)(2 133 24 93)(3 138 13 86)(4 143 14 91)(5 136 15 96)(6 141 16 89)(7 134 17 94)(8 139 18 87)(9 144 19 92)(10 137 20 85)(11 142 21 90)(12 135 22 95)(25 49 64 74)(26 54 65 79)(27 59 66 84)(28 52 67 77)(29 57 68 82)(30 50 69 75)(31 55 70 80)(32 60 71 73)(33 53 72 78)(34 58 61 83)(35 51 62 76)(36 56 63 81)(37 174 167 189)(38 179 168 182)(39 172 157 187)(40 177 158 192)(41 170 159 185)(42 175 160 190)(43 180 161 183)(44 173 162 188)(45 178 163 181)(46 171 164 186)(47 176 165 191)(48 169 166 184)(97 130 110 154)(98 123 111 147)(99 128 112 152)(100 121 113 145)(101 126 114 150)(102 131 115 155)(103 124 116 148)(104 129 117 153)(105 122 118 146)(106 127 119 151)(107 132 120 156)(108 125 109 149)
G:=sub<Sym(192)| (1,159,7,165)(2,42,8,48)(3,161,9,167)(4,44,10,38)(5,163,11,157)(6,46,12,40)(13,43,19,37)(14,162,20,168)(15,45,21,39)(16,164,22,158)(17,47,23,41)(18,166,24,160)(25,112,31,118)(26,100,32,106)(27,114,33,120)(28,102,34,108)(29,116,35,110)(30,104,36,98)(49,146,55,152)(50,123,56,129)(51,148,57,154)(52,125,58,131)(53,150,59,156)(54,127,60,121)(61,109,67,115)(62,97,68,103)(63,111,69,117)(64,99,70,105)(65,113,71,119)(66,101,72,107)(73,145,79,151)(74,122,80,128)(75,147,81,153)(76,124,82,130)(77,149,83,155)(78,126,84,132)(85,188,91,182)(86,174,92,180)(87,190,93,184)(88,176,94,170)(89,192,95,186)(90,178,96,172)(133,169,139,175)(134,185,140,191)(135,171,141,177)(136,187,142,181)(137,173,143,179)(138,189,144,183), (1,105,17,112)(2,113,18,106)(3,107,19,114)(4,115,20,108)(5,97,21,116)(6,117,22,98)(7,99,23,118)(8,119,24,100)(9,101,13,120)(10,109,14,102)(11,103,15,110)(12,111,16,104)(25,165,70,41)(26,42,71,166)(27,167,72,43)(28,44,61,168)(29,157,62,45)(30,46,63,158)(31,159,64,47)(32,48,65,160)(33,161,66,37)(34,38,67,162)(35,163,68,39)(36,40,69,164)(49,176,80,185)(50,186,81,177)(51,178,82,187)(52,188,83,179)(53,180,84,189)(54,190,73,169)(55,170,74,191)(56,192,75,171)(57,172,76,181)(58,182,77,173)(59,174,78,183)(60,184,79,175)(85,149,143,131)(86,132,144,150)(87,151,133,121)(88,122,134,152)(89,153,135,123)(90,124,136,154)(91,155,137,125)(92,126,138,156)(93,145,139,127)(94,128,140,146)(95,147,141,129)(96,130,142,148), (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), (1,140,23,88)(2,133,24,93)(3,138,13,86)(4,143,14,91)(5,136,15,96)(6,141,16,89)(7,134,17,94)(8,139,18,87)(9,144,19,92)(10,137,20,85)(11,142,21,90)(12,135,22,95)(25,49,64,74)(26,54,65,79)(27,59,66,84)(28,52,67,77)(29,57,68,82)(30,50,69,75)(31,55,70,80)(32,60,71,73)(33,53,72,78)(34,58,61,83)(35,51,62,76)(36,56,63,81)(37,174,167,189)(38,179,168,182)(39,172,157,187)(40,177,158,192)(41,170,159,185)(42,175,160,190)(43,180,161,183)(44,173,162,188)(45,178,163,181)(46,171,164,186)(47,176,165,191)(48,169,166,184)(97,130,110,154)(98,123,111,147)(99,128,112,152)(100,121,113,145)(101,126,114,150)(102,131,115,155)(103,124,116,148)(104,129,117,153)(105,122,118,146)(106,127,119,151)(107,132,120,156)(108,125,109,149)>;
G:=Group( (1,159,7,165)(2,42,8,48)(3,161,9,167)(4,44,10,38)(5,163,11,157)(6,46,12,40)(13,43,19,37)(14,162,20,168)(15,45,21,39)(16,164,22,158)(17,47,23,41)(18,166,24,160)(25,112,31,118)(26,100,32,106)(27,114,33,120)(28,102,34,108)(29,116,35,110)(30,104,36,98)(49,146,55,152)(50,123,56,129)(51,148,57,154)(52,125,58,131)(53,150,59,156)(54,127,60,121)(61,109,67,115)(62,97,68,103)(63,111,69,117)(64,99,70,105)(65,113,71,119)(66,101,72,107)(73,145,79,151)(74,122,80,128)(75,147,81,153)(76,124,82,130)(77,149,83,155)(78,126,84,132)(85,188,91,182)(86,174,92,180)(87,190,93,184)(88,176,94,170)(89,192,95,186)(90,178,96,172)(133,169,139,175)(134,185,140,191)(135,171,141,177)(136,187,142,181)(137,173,143,179)(138,189,144,183), (1,105,17,112)(2,113,18,106)(3,107,19,114)(4,115,20,108)(5,97,21,116)(6,117,22,98)(7,99,23,118)(8,119,24,100)(9,101,13,120)(10,109,14,102)(11,103,15,110)(12,111,16,104)(25,165,70,41)(26,42,71,166)(27,167,72,43)(28,44,61,168)(29,157,62,45)(30,46,63,158)(31,159,64,47)(32,48,65,160)(33,161,66,37)(34,38,67,162)(35,163,68,39)(36,40,69,164)(49,176,80,185)(50,186,81,177)(51,178,82,187)(52,188,83,179)(53,180,84,189)(54,190,73,169)(55,170,74,191)(56,192,75,171)(57,172,76,181)(58,182,77,173)(59,174,78,183)(60,184,79,175)(85,149,143,131)(86,132,144,150)(87,151,133,121)(88,122,134,152)(89,153,135,123)(90,124,136,154)(91,155,137,125)(92,126,138,156)(93,145,139,127)(94,128,140,146)(95,147,141,129)(96,130,142,148), (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), (1,140,23,88)(2,133,24,93)(3,138,13,86)(4,143,14,91)(5,136,15,96)(6,141,16,89)(7,134,17,94)(8,139,18,87)(9,144,19,92)(10,137,20,85)(11,142,21,90)(12,135,22,95)(25,49,64,74)(26,54,65,79)(27,59,66,84)(28,52,67,77)(29,57,68,82)(30,50,69,75)(31,55,70,80)(32,60,71,73)(33,53,72,78)(34,58,61,83)(35,51,62,76)(36,56,63,81)(37,174,167,189)(38,179,168,182)(39,172,157,187)(40,177,158,192)(41,170,159,185)(42,175,160,190)(43,180,161,183)(44,173,162,188)(45,178,163,181)(46,171,164,186)(47,176,165,191)(48,169,166,184)(97,130,110,154)(98,123,111,147)(99,128,112,152)(100,121,113,145)(101,126,114,150)(102,131,115,155)(103,124,116,148)(104,129,117,153)(105,122,118,146)(106,127,119,151)(107,132,120,156)(108,125,109,149) );
G=PermutationGroup([(1,159,7,165),(2,42,8,48),(3,161,9,167),(4,44,10,38),(5,163,11,157),(6,46,12,40),(13,43,19,37),(14,162,20,168),(15,45,21,39),(16,164,22,158),(17,47,23,41),(18,166,24,160),(25,112,31,118),(26,100,32,106),(27,114,33,120),(28,102,34,108),(29,116,35,110),(30,104,36,98),(49,146,55,152),(50,123,56,129),(51,148,57,154),(52,125,58,131),(53,150,59,156),(54,127,60,121),(61,109,67,115),(62,97,68,103),(63,111,69,117),(64,99,70,105),(65,113,71,119),(66,101,72,107),(73,145,79,151),(74,122,80,128),(75,147,81,153),(76,124,82,130),(77,149,83,155),(78,126,84,132),(85,188,91,182),(86,174,92,180),(87,190,93,184),(88,176,94,170),(89,192,95,186),(90,178,96,172),(133,169,139,175),(134,185,140,191),(135,171,141,177),(136,187,142,181),(137,173,143,179),(138,189,144,183)], [(1,105,17,112),(2,113,18,106),(3,107,19,114),(4,115,20,108),(5,97,21,116),(6,117,22,98),(7,99,23,118),(8,119,24,100),(9,101,13,120),(10,109,14,102),(11,103,15,110),(12,111,16,104),(25,165,70,41),(26,42,71,166),(27,167,72,43),(28,44,61,168),(29,157,62,45),(30,46,63,158),(31,159,64,47),(32,48,65,160),(33,161,66,37),(34,38,67,162),(35,163,68,39),(36,40,69,164),(49,176,80,185),(50,186,81,177),(51,178,82,187),(52,188,83,179),(53,180,84,189),(54,190,73,169),(55,170,74,191),(56,192,75,171),(57,172,76,181),(58,182,77,173),(59,174,78,183),(60,184,79,175),(85,149,143,131),(86,132,144,150),(87,151,133,121),(88,122,134,152),(89,153,135,123),(90,124,136,154),(91,155,137,125),(92,126,138,156),(93,145,139,127),(94,128,140,146),(95,147,141,129),(96,130,142,148)], [(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)], [(1,140,23,88),(2,133,24,93),(3,138,13,86),(4,143,14,91),(5,136,15,96),(6,141,16,89),(7,134,17,94),(8,139,18,87),(9,144,19,92),(10,137,20,85),(11,142,21,90),(12,135,22,95),(25,49,64,74),(26,54,65,79),(27,59,66,84),(28,52,67,77),(29,57,68,82),(30,50,69,75),(31,55,70,80),(32,60,71,73),(33,53,72,78),(34,58,61,83),(35,51,62,76),(36,56,63,81),(37,174,167,189),(38,179,168,182),(39,172,157,187),(40,177,158,192),(41,170,159,185),(42,175,160,190),(43,180,161,183),(44,173,162,188),(45,178,163,181),(46,171,164,186),(47,176,165,191),(48,169,166,184),(97,130,110,154),(98,123,111,147),(99,128,112,152),(100,121,113,145),(101,126,114,150),(102,131,115,155),(103,124,116,148),(104,129,117,153),(105,122,118,146),(106,127,119,151),(107,132,120,156),(108,125,109,149)])
Matrix representation ►G ⊆ 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] >;
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 | C12.3Q8 | C3×C42.C2 | C42.C2 | C42 | C4⋊C4 | C6 | C2 | C2 |
| # reps | 1 | 2 | 8 | 4 | 1 | 1 | 1 | 6 | 3 | 2 | 4 |
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
×
⋊
ℤ
𝔽
○
≀