metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.27D6, Dic3.2M4(2), C3⋊C8⋊9Q8, C4⋊C8.11S3, C3⋊3(C8⋊4Q8), C4.51(S3×Q8), C6.16(C4×Q8), (C2×C8).212D6, C24⋊C4.8C2, C6.11(C8○D4), C4⋊Dic3.11C4, C12.109(C2×Q8), (C2×Dic6).9C4, (C4×Dic6).7C2, C2.13(C8○D12), Dic3⋊C8.10C2, Dic3⋊C4.13C4, (C4×C12).54C22, (C8×Dic3).13C2, C2.15(S3×M4(2)), C6.24(C2×M4(2)), C12.301(C4○D4), (C2×C12).826C23, (C2×C24).248C22, C42.S3.2C2, C4.127(D4⋊2S3), C2.7(Dic6⋊C4), (C4×Dic3).273C22, (C3×C4⋊C8).19C2, (C2×C4).34(C4×S3), (C2×C12).42(C2×C4), C22.108(S3×C2×C4), (C2×C3⋊C8).193C22, (C2×C6).81(C22×C4), (C2×C4).768(C22×S3), (C2×Dic3).20(C2×C4), SmallGroup(192,387)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.27D6
G = < a,b,c,d | a4=b4=1, c6=b-1, d2=a2b, ab=ba, cac-1=a-1, dad-1=ab2, bc=cb, bd=db, dcd-1=b2c5 >
Subgroups: 184 in 94 conjugacy classes, 51 normal (47 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Dic3, Dic3, C12, C12, C2×C6, C42, C42, C4⋊C4, C2×C8, C2×C8, C2×Q8, C3⋊C8, C3⋊C8, C24, Dic6, C2×Dic3, C2×C12, C4×C8, C8⋊C4, C4⋊C8, C4⋊C8, C4×Q8, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4×C12, C2×C24, C2×Dic6, C8⋊4Q8, C42.S3, C8×Dic3, Dic3⋊C8, C24⋊C4, C3×C4⋊C8, C4×Dic6, C42.27D6
Quotients: C1, C2, C4, C22, S3, C2×C4, Q8, C23, D6, M4(2), C22×C4, C2×Q8, C4○D4, C4×S3, C22×S3, C4×Q8, C2×M4(2), C8○D4, S3×C2×C4, D4⋊2S3, S3×Q8, C8⋊4Q8, Dic6⋊C4, C8○D12, S3×M4(2), C42.27D6
►Regular action on 192 points
Generators in S192
(1 111 75 140)(2 141 76 112)(3 113 77 142)(4 143 78 114)(5 115 79 144)(6 121 80 116)(7 117 81 122)(8 123 82 118)(9 119 83 124)(10 125 84 120)(11 97 85 126)(12 127 86 98)(13 99 87 128)(14 129 88 100)(15 101 89 130)(16 131 90 102)(17 103 91 132)(18 133 92 104)(19 105 93 134)(20 135 94 106)(21 107 95 136)(22 137 96 108)(23 109 73 138)(24 139 74 110)(25 59 191 146)(26 147 192 60)(27 61 169 148)(28 149 170 62)(29 63 171 150)(30 151 172 64)(31 65 173 152)(32 153 174 66)(33 67 175 154)(34 155 176 68)(35 69 177 156)(36 157 178 70)(37 71 179 158)(38 159 180 72)(39 49 181 160)(40 161 182 50)(41 51 183 162)(42 163 184 52)(43 53 185 164)(44 165 186 54)(45 55 187 166)(46 167 188 56)(47 57 189 168)(48 145 190 58)
(1 19 13 7)(2 20 14 8)(3 21 15 9)(4 22 16 10)(5 23 17 11)(6 24 18 12)(25 43 37 31)(26 44 38 32)(27 45 39 33)(28 46 40 34)(29 47 41 35)(30 48 42 36)(49 67 61 55)(50 68 62 56)(51 69 63 57)(52 70 64 58)(53 71 65 59)(54 72 66 60)(73 91 85 79)(74 92 86 80)(75 93 87 81)(76 94 88 82)(77 95 89 83)(78 96 90 84)(97 115 109 103)(98 116 110 104)(99 117 111 105)(100 118 112 106)(101 119 113 107)(102 120 114 108)(121 139 133 127)(122 140 134 128)(123 141 135 129)(124 142 136 130)(125 143 137 131)(126 144 138 132)(145 163 157 151)(146 164 158 152)(147 165 159 153)(148 166 160 154)(149 167 161 155)(150 168 162 156)(169 187 181 175)(170 188 182 176)(171 189 183 177)(172 190 184 178)(173 191 185 179)(174 192 186 180)
(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 43 93 179 13 31 81 191)(2 36 94 172 14 48 82 184)(3 29 95 189 15 41 83 177)(4 46 96 182 16 34 84 170)(5 39 73 175 17 27 85 187)(6 32 74 192 18 44 86 180)(7 25 75 185 19 37 87 173)(8 42 76 178 20 30 88 190)(9 35 77 171 21 47 89 183)(10 28 78 188 22 40 90 176)(11 45 79 181 23 33 91 169)(12 38 80 174 24 26 92 186)(49 126 154 115 61 138 166 103)(50 143 155 108 62 131 167 120)(51 136 156 101 63 124 168 113)(52 129 157 118 64 141 145 106)(53 122 158 111 65 134 146 99)(54 139 159 104 66 127 147 116)(55 132 160 97 67 144 148 109)(56 125 161 114 68 137 149 102)(57 142 162 107 69 130 150 119)(58 135 163 100 70 123 151 112)(59 128 164 117 71 140 152 105)(60 121 165 110 72 133 153 98)
G:=sub<Sym(192)| (1,111,75,140)(2,141,76,112)(3,113,77,142)(4,143,78,114)(5,115,79,144)(6,121,80,116)(7,117,81,122)(8,123,82,118)(9,119,83,124)(10,125,84,120)(11,97,85,126)(12,127,86,98)(13,99,87,128)(14,129,88,100)(15,101,89,130)(16,131,90,102)(17,103,91,132)(18,133,92,104)(19,105,93,134)(20,135,94,106)(21,107,95,136)(22,137,96,108)(23,109,73,138)(24,139,74,110)(25,59,191,146)(26,147,192,60)(27,61,169,148)(28,149,170,62)(29,63,171,150)(30,151,172,64)(31,65,173,152)(32,153,174,66)(33,67,175,154)(34,155,176,68)(35,69,177,156)(36,157,178,70)(37,71,179,158)(38,159,180,72)(39,49,181,160)(40,161,182,50)(41,51,183,162)(42,163,184,52)(43,53,185,164)(44,165,186,54)(45,55,187,166)(46,167,188,56)(47,57,189,168)(48,145,190,58), (1,19,13,7)(2,20,14,8)(3,21,15,9)(4,22,16,10)(5,23,17,11)(6,24,18,12)(25,43,37,31)(26,44,38,32)(27,45,39,33)(28,46,40,34)(29,47,41,35)(30,48,42,36)(49,67,61,55)(50,68,62,56)(51,69,63,57)(52,70,64,58)(53,71,65,59)(54,72,66,60)(73,91,85,79)(74,92,86,80)(75,93,87,81)(76,94,88,82)(77,95,89,83)(78,96,90,84)(97,115,109,103)(98,116,110,104)(99,117,111,105)(100,118,112,106)(101,119,113,107)(102,120,114,108)(121,139,133,127)(122,140,134,128)(123,141,135,129)(124,142,136,130)(125,143,137,131)(126,144,138,132)(145,163,157,151)(146,164,158,152)(147,165,159,153)(148,166,160,154)(149,167,161,155)(150,168,162,156)(169,187,181,175)(170,188,182,176)(171,189,183,177)(172,190,184,178)(173,191,185,179)(174,192,186,180), (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), (1,43,93,179,13,31,81,191)(2,36,94,172,14,48,82,184)(3,29,95,189,15,41,83,177)(4,46,96,182,16,34,84,170)(5,39,73,175,17,27,85,187)(6,32,74,192,18,44,86,180)(7,25,75,185,19,37,87,173)(8,42,76,178,20,30,88,190)(9,35,77,171,21,47,89,183)(10,28,78,188,22,40,90,176)(11,45,79,181,23,33,91,169)(12,38,80,174,24,26,92,186)(49,126,154,115,61,138,166,103)(50,143,155,108,62,131,167,120)(51,136,156,101,63,124,168,113)(52,129,157,118,64,141,145,106)(53,122,158,111,65,134,146,99)(54,139,159,104,66,127,147,116)(55,132,160,97,67,144,148,109)(56,125,161,114,68,137,149,102)(57,142,162,107,69,130,150,119)(58,135,163,100,70,123,151,112)(59,128,164,117,71,140,152,105)(60,121,165,110,72,133,153,98)>;
G:=Group( (1,111,75,140)(2,141,76,112)(3,113,77,142)(4,143,78,114)(5,115,79,144)(6,121,80,116)(7,117,81,122)(8,123,82,118)(9,119,83,124)(10,125,84,120)(11,97,85,126)(12,127,86,98)(13,99,87,128)(14,129,88,100)(15,101,89,130)(16,131,90,102)(17,103,91,132)(18,133,92,104)(19,105,93,134)(20,135,94,106)(21,107,95,136)(22,137,96,108)(23,109,73,138)(24,139,74,110)(25,59,191,146)(26,147,192,60)(27,61,169,148)(28,149,170,62)(29,63,171,150)(30,151,172,64)(31,65,173,152)(32,153,174,66)(33,67,175,154)(34,155,176,68)(35,69,177,156)(36,157,178,70)(37,71,179,158)(38,159,180,72)(39,49,181,160)(40,161,182,50)(41,51,183,162)(42,163,184,52)(43,53,185,164)(44,165,186,54)(45,55,187,166)(46,167,188,56)(47,57,189,168)(48,145,190,58), (1,19,13,7)(2,20,14,8)(3,21,15,9)(4,22,16,10)(5,23,17,11)(6,24,18,12)(25,43,37,31)(26,44,38,32)(27,45,39,33)(28,46,40,34)(29,47,41,35)(30,48,42,36)(49,67,61,55)(50,68,62,56)(51,69,63,57)(52,70,64,58)(53,71,65,59)(54,72,66,60)(73,91,85,79)(74,92,86,80)(75,93,87,81)(76,94,88,82)(77,95,89,83)(78,96,90,84)(97,115,109,103)(98,116,110,104)(99,117,111,105)(100,118,112,106)(101,119,113,107)(102,120,114,108)(121,139,133,127)(122,140,134,128)(123,141,135,129)(124,142,136,130)(125,143,137,131)(126,144,138,132)(145,163,157,151)(146,164,158,152)(147,165,159,153)(148,166,160,154)(149,167,161,155)(150,168,162,156)(169,187,181,175)(170,188,182,176)(171,189,183,177)(172,190,184,178)(173,191,185,179)(174,192,186,180), (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), (1,43,93,179,13,31,81,191)(2,36,94,172,14,48,82,184)(3,29,95,189,15,41,83,177)(4,46,96,182,16,34,84,170)(5,39,73,175,17,27,85,187)(6,32,74,192,18,44,86,180)(7,25,75,185,19,37,87,173)(8,42,76,178,20,30,88,190)(9,35,77,171,21,47,89,183)(10,28,78,188,22,40,90,176)(11,45,79,181,23,33,91,169)(12,38,80,174,24,26,92,186)(49,126,154,115,61,138,166,103)(50,143,155,108,62,131,167,120)(51,136,156,101,63,124,168,113)(52,129,157,118,64,141,145,106)(53,122,158,111,65,134,146,99)(54,139,159,104,66,127,147,116)(55,132,160,97,67,144,148,109)(56,125,161,114,68,137,149,102)(57,142,162,107,69,130,150,119)(58,135,163,100,70,123,151,112)(59,128,164,117,71,140,152,105)(60,121,165,110,72,133,153,98) );
G=PermutationGroup([[(1,111,75,140),(2,141,76,112),(3,113,77,142),(4,143,78,114),(5,115,79,144),(6,121,80,116),(7,117,81,122),(8,123,82,118),(9,119,83,124),(10,125,84,120),(11,97,85,126),(12,127,86,98),(13,99,87,128),(14,129,88,100),(15,101,89,130),(16,131,90,102),(17,103,91,132),(18,133,92,104),(19,105,93,134),(20,135,94,106),(21,107,95,136),(22,137,96,108),(23,109,73,138),(24,139,74,110),(25,59,191,146),(26,147,192,60),(27,61,169,148),(28,149,170,62),(29,63,171,150),(30,151,172,64),(31,65,173,152),(32,153,174,66),(33,67,175,154),(34,155,176,68),(35,69,177,156),(36,157,178,70),(37,71,179,158),(38,159,180,72),(39,49,181,160),(40,161,182,50),(41,51,183,162),(42,163,184,52),(43,53,185,164),(44,165,186,54),(45,55,187,166),(46,167,188,56),(47,57,189,168),(48,145,190,58)], [(1,19,13,7),(2,20,14,8),(3,21,15,9),(4,22,16,10),(5,23,17,11),(6,24,18,12),(25,43,37,31),(26,44,38,32),(27,45,39,33),(28,46,40,34),(29,47,41,35),(30,48,42,36),(49,67,61,55),(50,68,62,56),(51,69,63,57),(52,70,64,58),(53,71,65,59),(54,72,66,60),(73,91,85,79),(74,92,86,80),(75,93,87,81),(76,94,88,82),(77,95,89,83),(78,96,90,84),(97,115,109,103),(98,116,110,104),(99,117,111,105),(100,118,112,106),(101,119,113,107),(102,120,114,108),(121,139,133,127),(122,140,134,128),(123,141,135,129),(124,142,136,130),(125,143,137,131),(126,144,138,132),(145,163,157,151),(146,164,158,152),(147,165,159,153),(148,166,160,154),(149,167,161,155),(150,168,162,156),(169,187,181,175),(170,188,182,176),(171,189,183,177),(172,190,184,178),(173,191,185,179),(174,192,186,180)], [(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)], [(1,43,93,179,13,31,81,191),(2,36,94,172,14,48,82,184),(3,29,95,189,15,41,83,177),(4,46,96,182,16,34,84,170),(5,39,73,175,17,27,85,187),(6,32,74,192,18,44,86,180),(7,25,75,185,19,37,87,173),(8,42,76,178,20,30,88,190),(9,35,77,171,21,47,89,183),(10,28,78,188,22,40,90,176),(11,45,79,181,23,33,91,169),(12,38,80,174,24,26,92,186),(49,126,154,115,61,138,166,103),(50,143,155,108,62,131,167,120),(51,136,156,101,63,124,168,113),(52,129,157,118,64,141,145,106),(53,122,158,111,65,134,146,99),(54,139,159,104,66,127,147,116),(55,132,160,97,67,144,148,109),(56,125,161,114,68,137,149,102),(57,142,162,107,69,130,150,119),(58,135,163,100,70,123,151,112),(59,128,164,117,71,140,152,105),(60,121,165,110,72,133,153,98)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 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 | C2 | C4 | C4 | C4 | S3 | Q8 | D6 | D6 | M4(2) | C4○D4 | C4×S3 | C8○D4 | C8○D12 | D4⋊2S3 | S3×Q8 | S3×M4(2) |
| kernel | C42.27D6 | C42.S3 | C8×Dic3 | Dic3⋊C8 | C24⋊C4 | C3×C4⋊C8 | C4×Dic6 | Dic3⋊C4 | C4⋊Dic3 | C2×Dic6 | C4⋊C8 | C3⋊C8 | C42 | C2×C8 | Dic3 | C12 | C2×C4 | C6 | C2 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 1 | 2 | 4 | 2 | 4 | 4 | 8 | 1 | 1 | 2 |
Matrix representation of C42.27D6 ►in GL4(𝔽73) generated by
| 43 | 13 | 0 | 0 |
| 60 | 30 | 0 | 0 |
| 0 | 0 | 14 | 59 |
| 0 | 0 | 61 | 59 |
| 46 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 22 | 22 | 0 | 0 |
| 51 | 0 | 0 | 0 |
| 0 | 0 | 63 | 3 |
| 0 | 0 | 40 | 10 |
| 36 | 5 | 0 | 0 |
| 42 | 37 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 46 |
G:=sub<GL(4,GF(73))| [43,60,0,0,13,30,0,0,0,0,14,61,0,0,59,59],[46,0,0,0,0,46,0,0,0,0,1,0,0,0,0,1],[22,51,0,0,22,0,0,0,0,0,63,40,0,0,3,10],[36,42,0,0,5,37,0,0,0,0,46,0,0,0,0,46] >;
C42.27D6 in GAP, Magma, Sage, TeX
C_4^2._{27}D_6 % in TeX
G:=Group("C4^2.27D6"); // GroupNames label
G:=SmallGroup(192,387);
// by ID
G=gap.SmallGroup(192,387);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,701,120,219,58,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=b^-1,d^2=a^2*b,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=b^2*c^5>;
// generators/relations