metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.198D6, C12.9M4(2), C3⋊C8⋊10Q8, C4⋊C8.12S3, C3⋊4(C8⋊4Q8), C6.18(C4×Q8), C4.53(S3×Q8), (C2×C8).180D6, C24⋊C4.9C2, C4.3(C8⋊S3), C6.27(C8○D4), C4⋊Dic3.17C4, C12.111(C2×Q8), C6.7(C2×M4(2)), Dic3⋊C8.12C2, Dic3⋊C4.12C4, (C4×C12).57C22, (C4×Dic6).10C2, (C2×Dic6).13C4, C12.303(C4○D4), (C2×C12).828C23, C2.12(D12.C4), (C2×C24).250C22, C4.129(D4⋊2S3), C2.8(Dic6⋊C4), (C4×Dic3).184C22, (C4×C3⋊C8).6C2, (C3×C4⋊C8).21C2, (C2×C4).71(C4×S3), C2.11(C2×C8⋊S3), C22.109(S3×C2×C4), (C2×C12).157(C2×C4), (C2×C3⋊C8).303C22, (C2×C6).83(C22×C4), (C2×C4).770(C22×S3), (C2×Dic3).21(C2×C4), SmallGroup(192,390)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.198D6
G = < a,b,c,d | a4=b4=1, c6=a2b-1, d2=a2, ab=ba, cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=c5 >
Subgroups: 184 in 94 conjugacy classes, 53 normal (31 characteristic)
C1, C2, C3, C4, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Dic3, C12, 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, C4×C3⋊C8, Dic3⋊C8, C24⋊C4, C3×C4⋊C8, C4×Dic6, C42.198D6
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, C8⋊S3, S3×C2×C4, D4⋊2S3, S3×Q8, C8⋊4Q8, Dic6⋊C4, C2×C8⋊S3, D12.C4, C42.198D6
►Regular action on 192 points
Generators in S192
(1 107 184 81)(2 82 185 108)(3 109 186 83)(4 84 187 110)(5 111 188 85)(6 86 189 112)(7 113 190 87)(8 88 191 114)(9 115 192 89)(10 90 169 116)(11 117 170 91)(12 92 171 118)(13 119 172 93)(14 94 173 120)(15 97 174 95)(16 96 175 98)(17 99 176 73)(18 74 177 100)(19 101 178 75)(20 76 179 102)(21 103 180 77)(22 78 181 104)(23 105 182 79)(24 80 183 106)(25 130 168 60)(26 61 145 131)(27 132 146 62)(28 63 147 133)(29 134 148 64)(30 65 149 135)(31 136 150 66)(32 67 151 137)(33 138 152 68)(34 69 153 139)(35 140 154 70)(36 71 155 141)(37 142 156 72)(38 49 157 143)(39 144 158 50)(40 51 159 121)(41 122 160 52)(42 53 161 123)(43 124 162 54)(44 55 163 125)(45 126 164 56)(46 57 165 127)(47 128 166 58)(48 59 167 129)
(1 178 13 190)(2 179 14 191)(3 180 15 192)(4 181 16 169)(5 182 17 170)(6 183 18 171)(7 184 19 172)(8 185 20 173)(9 186 21 174)(10 187 22 175)(11 188 23 176)(12 189 24 177)(25 162 37 150)(26 163 38 151)(27 164 39 152)(28 165 40 153)(29 166 41 154)(30 167 42 155)(31 168 43 156)(32 145 44 157)(33 146 45 158)(34 147 46 159)(35 148 47 160)(36 149 48 161)(49 137 61 125)(50 138 62 126)(51 139 63 127)(52 140 64 128)(53 141 65 129)(54 142 66 130)(55 143 67 131)(56 144 68 132)(57 121 69 133)(58 122 70 134)(59 123 71 135)(60 124 72 136)(73 117 85 105)(74 118 86 106)(75 119 87 107)(76 120 88 108)(77 97 89 109)(78 98 90 110)(79 99 91 111)(80 100 92 112)(81 101 93 113)(82 102 94 114)(83 103 95 115)(84 104 96 116)
(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)
(1 155 184 36)(2 160 185 41)(3 165 186 46)(4 146 187 27)(5 151 188 32)(6 156 189 37)(7 161 190 42)(8 166 191 47)(9 147 192 28)(10 152 169 33)(11 157 170 38)(12 162 171 43)(13 167 172 48)(14 148 173 29)(15 153 174 34)(16 158 175 39)(17 163 176 44)(18 168 177 25)(19 149 178 30)(20 154 179 35)(21 159 180 40)(22 164 181 45)(23 145 182 26)(24 150 183 31)(49 91 143 117)(50 96 144 98)(51 77 121 103)(52 82 122 108)(53 87 123 113)(54 92 124 118)(55 73 125 99)(56 78 126 104)(57 83 127 109)(58 88 128 114)(59 93 129 119)(60 74 130 100)(61 79 131 105)(62 84 132 110)(63 89 133 115)(64 94 134 120)(65 75 135 101)(66 80 136 106)(67 85 137 111)(68 90 138 116)(69 95 139 97)(70 76 140 102)(71 81 141 107)(72 86 142 112)
G:=sub<Sym(192)| (1,107,184,81)(2,82,185,108)(3,109,186,83)(4,84,187,110)(5,111,188,85)(6,86,189,112)(7,113,190,87)(8,88,191,114)(9,115,192,89)(10,90,169,116)(11,117,170,91)(12,92,171,118)(13,119,172,93)(14,94,173,120)(15,97,174,95)(16,96,175,98)(17,99,176,73)(18,74,177,100)(19,101,178,75)(20,76,179,102)(21,103,180,77)(22,78,181,104)(23,105,182,79)(24,80,183,106)(25,130,168,60)(26,61,145,131)(27,132,146,62)(28,63,147,133)(29,134,148,64)(30,65,149,135)(31,136,150,66)(32,67,151,137)(33,138,152,68)(34,69,153,139)(35,140,154,70)(36,71,155,141)(37,142,156,72)(38,49,157,143)(39,144,158,50)(40,51,159,121)(41,122,160,52)(42,53,161,123)(43,124,162,54)(44,55,163,125)(45,126,164,56)(46,57,165,127)(47,128,166,58)(48,59,167,129), (1,178,13,190)(2,179,14,191)(3,180,15,192)(4,181,16,169)(5,182,17,170)(6,183,18,171)(7,184,19,172)(8,185,20,173)(9,186,21,174)(10,187,22,175)(11,188,23,176)(12,189,24,177)(25,162,37,150)(26,163,38,151)(27,164,39,152)(28,165,40,153)(29,166,41,154)(30,167,42,155)(31,168,43,156)(32,145,44,157)(33,146,45,158)(34,147,46,159)(35,148,47,160)(36,149,48,161)(49,137,61,125)(50,138,62,126)(51,139,63,127)(52,140,64,128)(53,141,65,129)(54,142,66,130)(55,143,67,131)(56,144,68,132)(57,121,69,133)(58,122,70,134)(59,123,71,135)(60,124,72,136)(73,117,85,105)(74,118,86,106)(75,119,87,107)(76,120,88,108)(77,97,89,109)(78,98,90,110)(79,99,91,111)(80,100,92,112)(81,101,93,113)(82,102,94,114)(83,103,95,115)(84,104,96,116), (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), (1,155,184,36)(2,160,185,41)(3,165,186,46)(4,146,187,27)(5,151,188,32)(6,156,189,37)(7,161,190,42)(8,166,191,47)(9,147,192,28)(10,152,169,33)(11,157,170,38)(12,162,171,43)(13,167,172,48)(14,148,173,29)(15,153,174,34)(16,158,175,39)(17,163,176,44)(18,168,177,25)(19,149,178,30)(20,154,179,35)(21,159,180,40)(22,164,181,45)(23,145,182,26)(24,150,183,31)(49,91,143,117)(50,96,144,98)(51,77,121,103)(52,82,122,108)(53,87,123,113)(54,92,124,118)(55,73,125,99)(56,78,126,104)(57,83,127,109)(58,88,128,114)(59,93,129,119)(60,74,130,100)(61,79,131,105)(62,84,132,110)(63,89,133,115)(64,94,134,120)(65,75,135,101)(66,80,136,106)(67,85,137,111)(68,90,138,116)(69,95,139,97)(70,76,140,102)(71,81,141,107)(72,86,142,112)>;
G:=Group( (1,107,184,81)(2,82,185,108)(3,109,186,83)(4,84,187,110)(5,111,188,85)(6,86,189,112)(7,113,190,87)(8,88,191,114)(9,115,192,89)(10,90,169,116)(11,117,170,91)(12,92,171,118)(13,119,172,93)(14,94,173,120)(15,97,174,95)(16,96,175,98)(17,99,176,73)(18,74,177,100)(19,101,178,75)(20,76,179,102)(21,103,180,77)(22,78,181,104)(23,105,182,79)(24,80,183,106)(25,130,168,60)(26,61,145,131)(27,132,146,62)(28,63,147,133)(29,134,148,64)(30,65,149,135)(31,136,150,66)(32,67,151,137)(33,138,152,68)(34,69,153,139)(35,140,154,70)(36,71,155,141)(37,142,156,72)(38,49,157,143)(39,144,158,50)(40,51,159,121)(41,122,160,52)(42,53,161,123)(43,124,162,54)(44,55,163,125)(45,126,164,56)(46,57,165,127)(47,128,166,58)(48,59,167,129), (1,178,13,190)(2,179,14,191)(3,180,15,192)(4,181,16,169)(5,182,17,170)(6,183,18,171)(7,184,19,172)(8,185,20,173)(9,186,21,174)(10,187,22,175)(11,188,23,176)(12,189,24,177)(25,162,37,150)(26,163,38,151)(27,164,39,152)(28,165,40,153)(29,166,41,154)(30,167,42,155)(31,168,43,156)(32,145,44,157)(33,146,45,158)(34,147,46,159)(35,148,47,160)(36,149,48,161)(49,137,61,125)(50,138,62,126)(51,139,63,127)(52,140,64,128)(53,141,65,129)(54,142,66,130)(55,143,67,131)(56,144,68,132)(57,121,69,133)(58,122,70,134)(59,123,71,135)(60,124,72,136)(73,117,85,105)(74,118,86,106)(75,119,87,107)(76,120,88,108)(77,97,89,109)(78,98,90,110)(79,99,91,111)(80,100,92,112)(81,101,93,113)(82,102,94,114)(83,103,95,115)(84,104,96,116), (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), (1,155,184,36)(2,160,185,41)(3,165,186,46)(4,146,187,27)(5,151,188,32)(6,156,189,37)(7,161,190,42)(8,166,191,47)(9,147,192,28)(10,152,169,33)(11,157,170,38)(12,162,171,43)(13,167,172,48)(14,148,173,29)(15,153,174,34)(16,158,175,39)(17,163,176,44)(18,168,177,25)(19,149,178,30)(20,154,179,35)(21,159,180,40)(22,164,181,45)(23,145,182,26)(24,150,183,31)(49,91,143,117)(50,96,144,98)(51,77,121,103)(52,82,122,108)(53,87,123,113)(54,92,124,118)(55,73,125,99)(56,78,126,104)(57,83,127,109)(58,88,128,114)(59,93,129,119)(60,74,130,100)(61,79,131,105)(62,84,132,110)(63,89,133,115)(64,94,134,120)(65,75,135,101)(66,80,136,106)(67,85,137,111)(68,90,138,116)(69,95,139,97)(70,76,140,102)(71,81,141,107)(72,86,142,112) );
G=PermutationGroup([[(1,107,184,81),(2,82,185,108),(3,109,186,83),(4,84,187,110),(5,111,188,85),(6,86,189,112),(7,113,190,87),(8,88,191,114),(9,115,192,89),(10,90,169,116),(11,117,170,91),(12,92,171,118),(13,119,172,93),(14,94,173,120),(15,97,174,95),(16,96,175,98),(17,99,176,73),(18,74,177,100),(19,101,178,75),(20,76,179,102),(21,103,180,77),(22,78,181,104),(23,105,182,79),(24,80,183,106),(25,130,168,60),(26,61,145,131),(27,132,146,62),(28,63,147,133),(29,134,148,64),(30,65,149,135),(31,136,150,66),(32,67,151,137),(33,138,152,68),(34,69,153,139),(35,140,154,70),(36,71,155,141),(37,142,156,72),(38,49,157,143),(39,144,158,50),(40,51,159,121),(41,122,160,52),(42,53,161,123),(43,124,162,54),(44,55,163,125),(45,126,164,56),(46,57,165,127),(47,128,166,58),(48,59,167,129)], [(1,178,13,190),(2,179,14,191),(3,180,15,192),(4,181,16,169),(5,182,17,170),(6,183,18,171),(7,184,19,172),(8,185,20,173),(9,186,21,174),(10,187,22,175),(11,188,23,176),(12,189,24,177),(25,162,37,150),(26,163,38,151),(27,164,39,152),(28,165,40,153),(29,166,41,154),(30,167,42,155),(31,168,43,156),(32,145,44,157),(33,146,45,158),(34,147,46,159),(35,148,47,160),(36,149,48,161),(49,137,61,125),(50,138,62,126),(51,139,63,127),(52,140,64,128),(53,141,65,129),(54,142,66,130),(55,143,67,131),(56,144,68,132),(57,121,69,133),(58,122,70,134),(59,123,71,135),(60,124,72,136),(73,117,85,105),(74,118,86,106),(75,119,87,107),(76,120,88,108),(77,97,89,109),(78,98,90,110),(79,99,91,111),(80,100,92,112),(81,101,93,113),(82,102,94,114),(83,103,95,115),(84,104,96,116)], [(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)], [(1,155,184,36),(2,160,185,41),(3,165,186,46),(4,146,187,27),(5,151,188,32),(6,156,189,37),(7,161,190,42),(8,166,191,47),(9,147,192,28),(10,152,169,33),(11,157,170,38),(12,162,171,43),(13,167,172,48),(14,148,173,29),(15,153,174,34),(16,158,175,39),(17,163,176,44),(18,168,177,25),(19,149,178,30),(20,154,179,35),(21,159,180,40),(22,164,181,45),(23,145,182,26),(24,150,183,31),(49,91,143,117),(50,96,144,98),(51,77,121,103),(52,82,122,108),(53,87,123,113),(54,92,124,118),(55,73,125,99),(56,78,126,104),(57,83,127,109),(58,88,128,114),(59,93,129,119),(60,74,130,100),(61,79,131,105),(62,84,132,110),(63,89,133,115),(64,94,134,120),(65,75,135,101),(66,80,136,106),(67,85,137,111),(68,90,138,116),(69,95,139,97),(70,76,140,102),(71,81,141,107),(72,86,142,112)]])
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 | ··· | 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 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 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 | C4 | C4 | C4 | S3 | Q8 | D6 | D6 | M4(2) | C4○D4 | C4×S3 | C8○D4 | C8⋊S3 | D4⋊2S3 | S3×Q8 | D12.C4 |
| kernel | C42.198D6 | C4×C3⋊C8 | Dic3⋊C8 | C24⋊C4 | C3×C4⋊C8 | C4×Dic6 | Dic3⋊C4 | C4⋊Dic3 | C2×Dic6 | C4⋊C8 | C3⋊C8 | C42 | C2×C8 | C12 | C12 | C2×C4 | C6 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 1 | 2 | 4 | 2 | 4 | 4 | 8 | 1 | 1 | 2 |
Matrix representation of C42.198D6 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 61 | 46 |
| 0 | 0 | 27 | 12 |
| 46 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 70 | 3 | 0 | 0 |
| 70 | 67 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 3 | 31 | 0 | 0 |
| 28 | 70 | 0 | 0 |
| 0 | 0 | 0 | 46 |
| 0 | 0 | 46 | 0 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,61,27,0,0,46,12],[46,0,0,0,0,46,0,0,0,0,72,0,0,0,0,72],[70,70,0,0,3,67,0,0,0,0,0,1,0,0,1,0],[3,28,0,0,31,70,0,0,0,0,0,46,0,0,46,0] >;
C42.198D6 in GAP, Magma, Sage, TeX
C_4^2._{198}D_6 % in TeX
G:=Group("C4^2.198D6"); // GroupNames label
G:=SmallGroup(192,390);
// by ID
G=gap.SmallGroup(192,390);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,64,758,135,142,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=a^2*b^-1,d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^5>;
// generators/relations