metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊6Dic6, C42.123D6, C6.82- (1+4), (C3×Q8)⋊6Q8, C4⋊C4.292D6, C3⋊2(Q8⋊3Q8), C12⋊Q8.11C2, (C4×Q8).18S3, C12.45(C2×Q8), (Q8×C12).12C2, (C2×Q8).221D6, C4.18(C2×Dic6), C6.16(C22×Q8), (C2×C6).113C24, (C4×Dic6).22C2, (Q8×Dic3).11C2, (C4×C12).165C22, (C2×C12).493C23, Dic3⋊C4.9C22, C12.6Q8.11C2, C12.3Q8.10C2, (C6×Q8).213C22, C2.18(C22×Dic6), Dic3.36(C4○D4), C4⋊Dic3.304C22, C22.138(S3×C23), (C2×Dic3).51C23, (C4×Dic3).82C22, C2.11(Q8.15D6), (C2×Dic6).241C22, C2.28(S3×C4○D4), C6.143(C2×C4○D4), (C3×C4⋊C4).341C22, (C2×C4).167(C22×S3), SmallGroup(192,1128)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 392 in 200 conjugacy classes, 113 normal (22 characteristic)
C1, C2 [×3], C3, C4 [×6], C4 [×13], C22, C6 [×3], C2×C4, C2×C4 [×6], C2×C4 [×8], Q8 [×4], Q8 [×6], Dic3 [×2], Dic3 [×7], C12 [×6], C12 [×4], C2×C6, C42 [×3], C42 [×6], C4⋊C4 [×3], C4⋊C4 [×19], C2×Q8, C2×Q8 [×3], Dic6 [×6], C2×Dic3 [×2], C2×Dic3 [×6], C2×C12, C2×C12 [×6], C3×Q8 [×4], C4×Q8, C4×Q8 [×5], C42.C2 [×6], C4⋊Q8 [×3], C4×Dic3 [×6], Dic3⋊C4, Dic3⋊C4 [×9], C4⋊Dic3 [×9], C4×C12 [×3], C3×C4⋊C4 [×3], C2×Dic6 [×3], C6×Q8, Q8⋊3Q8, C4×Dic6 [×3], C12.6Q8 [×3], C12⋊Q8 [×3], C12.3Q8 [×3], Q8×Dic3 [×2], Q8×C12, Q8⋊6Dic6
Quotients:
C1, C2 [×15], C22 [×35], S3, Q8 [×4], C23 [×15], D6 [×7], C2×Q8 [×6], C4○D4 [×2], C24, Dic6 [×4], C22×S3 [×7], C22×Q8, C2×C4○D4, 2- (1+4), C2×Dic6 [×6], S3×C23, Q8⋊3Q8, C22×Dic6, Q8.15D6, S3×C4○D4, Q8⋊6Dic6
Generators and relations
G = < a,b,c,d | a4=c12=1, b2=a2, d2=c6, bab-1=a-1, ac=ca, ad=da, cbc-1=a2b, bd=db, dcd-1=c-1 >
►Regular action on 192 points
Generators in S192
(1 139 51 23)(2 140 52 24)(3 141 53 13)(4 142 54 14)(5 143 55 15)(6 144 56 16)(7 133 57 17)(8 134 58 18)(9 135 59 19)(10 136 60 20)(11 137 49 21)(12 138 50 22)(25 188 79 132)(26 189 80 121)(27 190 81 122)(28 191 82 123)(29 192 83 124)(30 181 84 125)(31 182 73 126)(32 183 74 127)(33 184 75 128)(34 185 76 129)(35 186 77 130)(36 187 78 131)(37 149 98 175)(38 150 99 176)(39 151 100 177)(40 152 101 178)(41 153 102 179)(42 154 103 180)(43 155 104 169)(44 156 105 170)(45 145 106 171)(46 146 107 172)(47 147 108 173)(48 148 97 174)(61 166 111 96)(62 167 112 85)(63 168 113 86)(64 157 114 87)(65 158 115 88)(66 159 116 89)(67 160 117 90)(68 161 118 91)(69 162 119 92)(70 163 120 93)(71 164 109 94)(72 165 110 95)
(1 85 51 167)(2 168 52 86)(3 87 53 157)(4 158 54 88)(5 89 55 159)(6 160 56 90)(7 91 57 161)(8 162 58 92)(9 93 59 163)(10 164 60 94)(11 95 49 165)(12 166 50 96)(13 64 141 114)(14 115 142 65)(15 66 143 116)(16 117 144 67)(17 68 133 118)(18 119 134 69)(19 70 135 120)(20 109 136 71)(21 72 137 110)(22 111 138 61)(23 62 139 112)(24 113 140 63)(25 38 79 99)(26 100 80 39)(27 40 81 101)(28 102 82 41)(29 42 83 103)(30 104 84 43)(31 44 73 105)(32 106 74 45)(33 46 75 107)(34 108 76 47)(35 48 77 97)(36 98 78 37)(121 177 189 151)(122 152 190 178)(123 179 191 153)(124 154 192 180)(125 169 181 155)(126 156 182 170)(127 171 183 145)(128 146 184 172)(129 173 185 147)(130 148 186 174)(131 175 187 149)(132 150 188 176)
(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 42 7 48)(2 41 8 47)(3 40 9 46)(4 39 10 45)(5 38 11 44)(6 37 12 43)(13 178 19 172)(14 177 20 171)(15 176 21 170)(16 175 22 169)(17 174 23 180)(18 173 24 179)(25 165 31 159)(26 164 32 158)(27 163 33 157)(28 162 34 168)(29 161 35 167)(30 160 36 166)(49 105 55 99)(50 104 56 98)(51 103 57 97)(52 102 58 108)(53 101 59 107)(54 100 60 106)(61 125 67 131)(62 124 68 130)(63 123 69 129)(64 122 70 128)(65 121 71 127)(66 132 72 126)(73 89 79 95)(74 88 80 94)(75 87 81 93)(76 86 82 92)(77 85 83 91)(78 96 84 90)(109 183 115 189)(110 182 116 188)(111 181 117 187)(112 192 118 186)(113 191 119 185)(114 190 120 184)(133 148 139 154)(134 147 140 153)(135 146 141 152)(136 145 142 151)(137 156 143 150)(138 155 144 149)
G:=sub<Sym(192)| (1,139,51,23)(2,140,52,24)(3,141,53,13)(4,142,54,14)(5,143,55,15)(6,144,56,16)(7,133,57,17)(8,134,58,18)(9,135,59,19)(10,136,60,20)(11,137,49,21)(12,138,50,22)(25,188,79,132)(26,189,80,121)(27,190,81,122)(28,191,82,123)(29,192,83,124)(30,181,84,125)(31,182,73,126)(32,183,74,127)(33,184,75,128)(34,185,76,129)(35,186,77,130)(36,187,78,131)(37,149,98,175)(38,150,99,176)(39,151,100,177)(40,152,101,178)(41,153,102,179)(42,154,103,180)(43,155,104,169)(44,156,105,170)(45,145,106,171)(46,146,107,172)(47,147,108,173)(48,148,97,174)(61,166,111,96)(62,167,112,85)(63,168,113,86)(64,157,114,87)(65,158,115,88)(66,159,116,89)(67,160,117,90)(68,161,118,91)(69,162,119,92)(70,163,120,93)(71,164,109,94)(72,165,110,95), (1,85,51,167)(2,168,52,86)(3,87,53,157)(4,158,54,88)(5,89,55,159)(6,160,56,90)(7,91,57,161)(8,162,58,92)(9,93,59,163)(10,164,60,94)(11,95,49,165)(12,166,50,96)(13,64,141,114)(14,115,142,65)(15,66,143,116)(16,117,144,67)(17,68,133,118)(18,119,134,69)(19,70,135,120)(20,109,136,71)(21,72,137,110)(22,111,138,61)(23,62,139,112)(24,113,140,63)(25,38,79,99)(26,100,80,39)(27,40,81,101)(28,102,82,41)(29,42,83,103)(30,104,84,43)(31,44,73,105)(32,106,74,45)(33,46,75,107)(34,108,76,47)(35,48,77,97)(36,98,78,37)(121,177,189,151)(122,152,190,178)(123,179,191,153)(124,154,192,180)(125,169,181,155)(126,156,182,170)(127,171,183,145)(128,146,184,172)(129,173,185,147)(130,148,186,174)(131,175,187,149)(132,150,188,176), (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), (1,42,7,48)(2,41,8,47)(3,40,9,46)(4,39,10,45)(5,38,11,44)(6,37,12,43)(13,178,19,172)(14,177,20,171)(15,176,21,170)(16,175,22,169)(17,174,23,180)(18,173,24,179)(25,165,31,159)(26,164,32,158)(27,163,33,157)(28,162,34,168)(29,161,35,167)(30,160,36,166)(49,105,55,99)(50,104,56,98)(51,103,57,97)(52,102,58,108)(53,101,59,107)(54,100,60,106)(61,125,67,131)(62,124,68,130)(63,123,69,129)(64,122,70,128)(65,121,71,127)(66,132,72,126)(73,89,79,95)(74,88,80,94)(75,87,81,93)(76,86,82,92)(77,85,83,91)(78,96,84,90)(109,183,115,189)(110,182,116,188)(111,181,117,187)(112,192,118,186)(113,191,119,185)(114,190,120,184)(133,148,139,154)(134,147,140,153)(135,146,141,152)(136,145,142,151)(137,156,143,150)(138,155,144,149)>;
G:=Group( (1,139,51,23)(2,140,52,24)(3,141,53,13)(4,142,54,14)(5,143,55,15)(6,144,56,16)(7,133,57,17)(8,134,58,18)(9,135,59,19)(10,136,60,20)(11,137,49,21)(12,138,50,22)(25,188,79,132)(26,189,80,121)(27,190,81,122)(28,191,82,123)(29,192,83,124)(30,181,84,125)(31,182,73,126)(32,183,74,127)(33,184,75,128)(34,185,76,129)(35,186,77,130)(36,187,78,131)(37,149,98,175)(38,150,99,176)(39,151,100,177)(40,152,101,178)(41,153,102,179)(42,154,103,180)(43,155,104,169)(44,156,105,170)(45,145,106,171)(46,146,107,172)(47,147,108,173)(48,148,97,174)(61,166,111,96)(62,167,112,85)(63,168,113,86)(64,157,114,87)(65,158,115,88)(66,159,116,89)(67,160,117,90)(68,161,118,91)(69,162,119,92)(70,163,120,93)(71,164,109,94)(72,165,110,95), (1,85,51,167)(2,168,52,86)(3,87,53,157)(4,158,54,88)(5,89,55,159)(6,160,56,90)(7,91,57,161)(8,162,58,92)(9,93,59,163)(10,164,60,94)(11,95,49,165)(12,166,50,96)(13,64,141,114)(14,115,142,65)(15,66,143,116)(16,117,144,67)(17,68,133,118)(18,119,134,69)(19,70,135,120)(20,109,136,71)(21,72,137,110)(22,111,138,61)(23,62,139,112)(24,113,140,63)(25,38,79,99)(26,100,80,39)(27,40,81,101)(28,102,82,41)(29,42,83,103)(30,104,84,43)(31,44,73,105)(32,106,74,45)(33,46,75,107)(34,108,76,47)(35,48,77,97)(36,98,78,37)(121,177,189,151)(122,152,190,178)(123,179,191,153)(124,154,192,180)(125,169,181,155)(126,156,182,170)(127,171,183,145)(128,146,184,172)(129,173,185,147)(130,148,186,174)(131,175,187,149)(132,150,188,176), (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), (1,42,7,48)(2,41,8,47)(3,40,9,46)(4,39,10,45)(5,38,11,44)(6,37,12,43)(13,178,19,172)(14,177,20,171)(15,176,21,170)(16,175,22,169)(17,174,23,180)(18,173,24,179)(25,165,31,159)(26,164,32,158)(27,163,33,157)(28,162,34,168)(29,161,35,167)(30,160,36,166)(49,105,55,99)(50,104,56,98)(51,103,57,97)(52,102,58,108)(53,101,59,107)(54,100,60,106)(61,125,67,131)(62,124,68,130)(63,123,69,129)(64,122,70,128)(65,121,71,127)(66,132,72,126)(73,89,79,95)(74,88,80,94)(75,87,81,93)(76,86,82,92)(77,85,83,91)(78,96,84,90)(109,183,115,189)(110,182,116,188)(111,181,117,187)(112,192,118,186)(113,191,119,185)(114,190,120,184)(133,148,139,154)(134,147,140,153)(135,146,141,152)(136,145,142,151)(137,156,143,150)(138,155,144,149) );
G=PermutationGroup([(1,139,51,23),(2,140,52,24),(3,141,53,13),(4,142,54,14),(5,143,55,15),(6,144,56,16),(7,133,57,17),(8,134,58,18),(9,135,59,19),(10,136,60,20),(11,137,49,21),(12,138,50,22),(25,188,79,132),(26,189,80,121),(27,190,81,122),(28,191,82,123),(29,192,83,124),(30,181,84,125),(31,182,73,126),(32,183,74,127),(33,184,75,128),(34,185,76,129),(35,186,77,130),(36,187,78,131),(37,149,98,175),(38,150,99,176),(39,151,100,177),(40,152,101,178),(41,153,102,179),(42,154,103,180),(43,155,104,169),(44,156,105,170),(45,145,106,171),(46,146,107,172),(47,147,108,173),(48,148,97,174),(61,166,111,96),(62,167,112,85),(63,168,113,86),(64,157,114,87),(65,158,115,88),(66,159,116,89),(67,160,117,90),(68,161,118,91),(69,162,119,92),(70,163,120,93),(71,164,109,94),(72,165,110,95)], [(1,85,51,167),(2,168,52,86),(3,87,53,157),(4,158,54,88),(5,89,55,159),(6,160,56,90),(7,91,57,161),(8,162,58,92),(9,93,59,163),(10,164,60,94),(11,95,49,165),(12,166,50,96),(13,64,141,114),(14,115,142,65),(15,66,143,116),(16,117,144,67),(17,68,133,118),(18,119,134,69),(19,70,135,120),(20,109,136,71),(21,72,137,110),(22,111,138,61),(23,62,139,112),(24,113,140,63),(25,38,79,99),(26,100,80,39),(27,40,81,101),(28,102,82,41),(29,42,83,103),(30,104,84,43),(31,44,73,105),(32,106,74,45),(33,46,75,107),(34,108,76,47),(35,48,77,97),(36,98,78,37),(121,177,189,151),(122,152,190,178),(123,179,191,153),(124,154,192,180),(125,169,181,155),(126,156,182,170),(127,171,183,145),(128,146,184,172),(129,173,185,147),(130,148,186,174),(131,175,187,149),(132,150,188,176)], [(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)], [(1,42,7,48),(2,41,8,47),(3,40,9,46),(4,39,10,45),(5,38,11,44),(6,37,12,43),(13,178,19,172),(14,177,20,171),(15,176,21,170),(16,175,22,169),(17,174,23,180),(18,173,24,179),(25,165,31,159),(26,164,32,158),(27,163,33,157),(28,162,34,168),(29,161,35,167),(30,160,36,166),(49,105,55,99),(50,104,56,98),(51,103,57,97),(52,102,58,108),(53,101,59,107),(54,100,60,106),(61,125,67,131),(62,124,68,130),(63,123,69,129),(64,122,70,128),(65,121,71,127),(66,132,72,126),(73,89,79,95),(74,88,80,94),(75,87,81,93),(76,86,82,92),(77,85,83,91),(78,96,84,90),(109,183,115,189),(110,182,116,188),(111,181,117,187),(112,192,118,186),(113,191,119,185),(114,190,120,184),(133,148,139,154),(134,147,140,153),(135,146,141,152),(136,145,142,151),(137,156,143,150),(138,155,144,149)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 10 |
| 0 | 0 | 10 | 4 |
| 2 | 0 | 0 | 0 |
| 2 | 7 | 0 | 0 |
| 0 | 0 | 0 | 5 |
| 0 | 0 | 8 | 0 |
| 6 | 2 | 0 | 0 |
| 1 | 7 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,0,12,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,9,10,0,0,10,4],[2,2,0,0,0,7,0,0,0,0,0,8,0,0,5,0],[6,1,0,0,2,7,0,0,0,0,1,0,0,0,0,1] >;
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | ··· | 4U | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | D6 | C4○D4 | Dic6 | 2- (1+4) | Q8.15D6 | S3×C4○D4 |
| kernel | Q8⋊6Dic6 | C4×Dic6 | C12.6Q8 | C12⋊Q8 | C12.3Q8 | Q8×Dic3 | Q8×C12 | C4×Q8 | C3×Q8 | C42 | C4⋊C4 | C2×Q8 | Dic3 | Q8 | C6 | C2 | C2 |
| # reps | 1 | 3 | 3 | 3 | 3 | 2 | 1 | 1 | 4 | 3 | 3 | 1 | 4 | 8 | 1 | 2 | 2 |
In GAP, Magma, Sage, TeX
Q_8\rtimes_6Dic_6
% in TeX
G:=Group("Q8:6Dic6"); // GroupNames label
G:=SmallGroup(192,1128);
// by ID
G=gap.SmallGroup(192,1128);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,232,387,184,675,192,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^12=1,b^2=a^2,d^2=c^6,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀