metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊4Dic6, C42.53D6, C12.48SD16, (C3×Q8)⋊3Q8, (C4×Q8).9S3, C4⋊C4.248D6, C3⋊4(Q8⋊Q8), (C2×C12).63D4, (Q8×C12).2C2, C12.28(C2×Q8), (C2×Q8).179D6, C12⋊C8.14C2, C4.12(C2×Dic6), C6.68(C2×SD16), C12.55(C4○D4), C4.62(C4○D12), (C4×C12).91C22, C12⋊2Q8.13C2, Q8⋊2Dic3.7C2, C6.64(C22⋊Q8), (C2×C12).342C23, C4.13(Q8⋊2S3), C2.9(Q8.14D6), C12.Q8.10C2, (C6×Q8).190C22, C6.110(C8.C22), C4⋊Dic3.139C22, C2.15(C12.48D4), (C2×C6).473(C2×D4), (C2×C3⋊C8).97C22, C2.6(C2×Q8⋊2S3), (C2×C4).247(C3⋊D4), (C3×C4⋊C4).279C22, (C2×C4).442(C22×S3), C22.152(C2×C3⋊D4), SmallGroup(192,579)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊4Dic6
G = < a,b,c,d | a4=c12=1, b2=a2, d2=c6, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=c-1 >
Subgroups: 232 in 96 conjugacy classes, 47 normal (31 characteristic)
C1, C2, C3, C4, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Q8, Dic3, C12, C12, C12, C2×C6, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C2×Q8, C3⋊C8, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Q8, Q8⋊C4, C4⋊C8, C4.Q8, C4×Q8, C4⋊Q8, C2×C3⋊C8, C4⋊Dic3, C4⋊Dic3, C4×C12, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C6×Q8, Q8⋊Q8, C12⋊C8, C12.Q8, Q8⋊2Dic3, C12⋊2Q8, Q8×C12, Q8⋊4Dic6
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, SD16, C2×D4, C2×Q8, C4○D4, Dic6, C3⋊D4, C22×S3, C22⋊Q8, C2×SD16, C8.C22, Q8⋊2S3, C2×Dic6, C4○D12, C2×C3⋊D4, Q8⋊Q8, C12.48D4, C2×Q8⋊2S3, Q8.14D6, Q8⋊4Dic6
►Regular action on 192 points
Generators in S192
(1 162 150 172)(2 163 151 173)(3 164 152 174)(4 165 153 175)(5 166 154 176)(6 167 155 177)(7 168 156 178)(8 157 145 179)(9 158 146 180)(10 159 147 169)(11 160 148 170)(12 161 149 171)(13 75 142 113)(14 76 143 114)(15 77 144 115)(16 78 133 116)(17 79 134 117)(18 80 135 118)(19 81 136 119)(20 82 137 120)(21 83 138 109)(22 84 139 110)(23 73 140 111)(24 74 141 112)(25 57 132 63)(26 58 121 64)(27 59 122 65)(28 60 123 66)(29 49 124 67)(30 50 125 68)(31 51 126 69)(32 52 127 70)(33 53 128 71)(34 54 129 72)(35 55 130 61)(36 56 131 62)(37 183 105 88)(38 184 106 89)(39 185 107 90)(40 186 108 91)(41 187 97 92)(42 188 98 93)(43 189 99 94)(44 190 100 95)(45 191 101 96)(46 192 102 85)(47 181 103 86)(48 182 104 87)
(1 186 150 91)(2 187 151 92)(3 188 152 93)(4 189 153 94)(5 190 154 95)(6 191 155 96)(7 192 156 85)(8 181 145 86)(9 182 146 87)(10 183 147 88)(11 184 148 89)(12 185 149 90)(13 30 142 125)(14 31 143 126)(15 32 144 127)(16 33 133 128)(17 34 134 129)(18 35 135 130)(19 36 136 131)(20 25 137 132)(21 26 138 121)(22 27 139 122)(23 28 140 123)(24 29 141 124)(37 169 105 159)(38 170 106 160)(39 171 107 161)(40 172 108 162)(41 173 97 163)(42 174 98 164)(43 175 99 165)(44 176 100 166)(45 177 101 167)(46 178 102 168)(47 179 103 157)(48 180 104 158)(49 74 67 112)(50 75 68 113)(51 76 69 114)(52 77 70 115)(53 78 71 116)(54 79 72 117)(55 80 61 118)(56 81 62 119)(57 82 63 120)(58 83 64 109)(59 84 65 110)(60 73 66 111)
(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)
(1 142 7 136)(2 141 8 135)(3 140 9 134)(4 139 10 133)(5 138 11 144)(6 137 12 143)(13 156 19 150)(14 155 20 149)(15 154 21 148)(16 153 22 147)(17 152 23 146)(18 151 24 145)(25 39 31 45)(26 38 32 44)(27 37 33 43)(28 48 34 42)(29 47 35 41)(30 46 36 40)(49 86 55 92)(50 85 56 91)(51 96 57 90)(52 95 58 89)(53 94 59 88)(54 93 60 87)(61 187 67 181)(62 186 68 192)(63 185 69 191)(64 184 70 190)(65 183 71 189)(66 182 72 188)(73 158 79 164)(74 157 80 163)(75 168 81 162)(76 167 82 161)(77 166 83 160)(78 165 84 159)(97 124 103 130)(98 123 104 129)(99 122 105 128)(100 121 106 127)(101 132 107 126)(102 131 108 125)(109 170 115 176)(110 169 116 175)(111 180 117 174)(112 179 118 173)(113 178 119 172)(114 177 120 171)
G:=sub<Sym(192)| (1,162,150,172)(2,163,151,173)(3,164,152,174)(4,165,153,175)(5,166,154,176)(6,167,155,177)(7,168,156,178)(8,157,145,179)(9,158,146,180)(10,159,147,169)(11,160,148,170)(12,161,149,171)(13,75,142,113)(14,76,143,114)(15,77,144,115)(16,78,133,116)(17,79,134,117)(18,80,135,118)(19,81,136,119)(20,82,137,120)(21,83,138,109)(22,84,139,110)(23,73,140,111)(24,74,141,112)(25,57,132,63)(26,58,121,64)(27,59,122,65)(28,60,123,66)(29,49,124,67)(30,50,125,68)(31,51,126,69)(32,52,127,70)(33,53,128,71)(34,54,129,72)(35,55,130,61)(36,56,131,62)(37,183,105,88)(38,184,106,89)(39,185,107,90)(40,186,108,91)(41,187,97,92)(42,188,98,93)(43,189,99,94)(44,190,100,95)(45,191,101,96)(46,192,102,85)(47,181,103,86)(48,182,104,87), (1,186,150,91)(2,187,151,92)(3,188,152,93)(4,189,153,94)(5,190,154,95)(6,191,155,96)(7,192,156,85)(8,181,145,86)(9,182,146,87)(10,183,147,88)(11,184,148,89)(12,185,149,90)(13,30,142,125)(14,31,143,126)(15,32,144,127)(16,33,133,128)(17,34,134,129)(18,35,135,130)(19,36,136,131)(20,25,137,132)(21,26,138,121)(22,27,139,122)(23,28,140,123)(24,29,141,124)(37,169,105,159)(38,170,106,160)(39,171,107,161)(40,172,108,162)(41,173,97,163)(42,174,98,164)(43,175,99,165)(44,176,100,166)(45,177,101,167)(46,178,102,168)(47,179,103,157)(48,180,104,158)(49,74,67,112)(50,75,68,113)(51,76,69,114)(52,77,70,115)(53,78,71,116)(54,79,72,117)(55,80,61,118)(56,81,62,119)(57,82,63,120)(58,83,64,109)(59,84,65,110)(60,73,66,111), (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), (1,142,7,136)(2,141,8,135)(3,140,9,134)(4,139,10,133)(5,138,11,144)(6,137,12,143)(13,156,19,150)(14,155,20,149)(15,154,21,148)(16,153,22,147)(17,152,23,146)(18,151,24,145)(25,39,31,45)(26,38,32,44)(27,37,33,43)(28,48,34,42)(29,47,35,41)(30,46,36,40)(49,86,55,92)(50,85,56,91)(51,96,57,90)(52,95,58,89)(53,94,59,88)(54,93,60,87)(61,187,67,181)(62,186,68,192)(63,185,69,191)(64,184,70,190)(65,183,71,189)(66,182,72,188)(73,158,79,164)(74,157,80,163)(75,168,81,162)(76,167,82,161)(77,166,83,160)(78,165,84,159)(97,124,103,130)(98,123,104,129)(99,122,105,128)(100,121,106,127)(101,132,107,126)(102,131,108,125)(109,170,115,176)(110,169,116,175)(111,180,117,174)(112,179,118,173)(113,178,119,172)(114,177,120,171)>;
G:=Group( (1,162,150,172)(2,163,151,173)(3,164,152,174)(4,165,153,175)(5,166,154,176)(6,167,155,177)(7,168,156,178)(8,157,145,179)(9,158,146,180)(10,159,147,169)(11,160,148,170)(12,161,149,171)(13,75,142,113)(14,76,143,114)(15,77,144,115)(16,78,133,116)(17,79,134,117)(18,80,135,118)(19,81,136,119)(20,82,137,120)(21,83,138,109)(22,84,139,110)(23,73,140,111)(24,74,141,112)(25,57,132,63)(26,58,121,64)(27,59,122,65)(28,60,123,66)(29,49,124,67)(30,50,125,68)(31,51,126,69)(32,52,127,70)(33,53,128,71)(34,54,129,72)(35,55,130,61)(36,56,131,62)(37,183,105,88)(38,184,106,89)(39,185,107,90)(40,186,108,91)(41,187,97,92)(42,188,98,93)(43,189,99,94)(44,190,100,95)(45,191,101,96)(46,192,102,85)(47,181,103,86)(48,182,104,87), (1,186,150,91)(2,187,151,92)(3,188,152,93)(4,189,153,94)(5,190,154,95)(6,191,155,96)(7,192,156,85)(8,181,145,86)(9,182,146,87)(10,183,147,88)(11,184,148,89)(12,185,149,90)(13,30,142,125)(14,31,143,126)(15,32,144,127)(16,33,133,128)(17,34,134,129)(18,35,135,130)(19,36,136,131)(20,25,137,132)(21,26,138,121)(22,27,139,122)(23,28,140,123)(24,29,141,124)(37,169,105,159)(38,170,106,160)(39,171,107,161)(40,172,108,162)(41,173,97,163)(42,174,98,164)(43,175,99,165)(44,176,100,166)(45,177,101,167)(46,178,102,168)(47,179,103,157)(48,180,104,158)(49,74,67,112)(50,75,68,113)(51,76,69,114)(52,77,70,115)(53,78,71,116)(54,79,72,117)(55,80,61,118)(56,81,62,119)(57,82,63,120)(58,83,64,109)(59,84,65,110)(60,73,66,111), (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), (1,142,7,136)(2,141,8,135)(3,140,9,134)(4,139,10,133)(5,138,11,144)(6,137,12,143)(13,156,19,150)(14,155,20,149)(15,154,21,148)(16,153,22,147)(17,152,23,146)(18,151,24,145)(25,39,31,45)(26,38,32,44)(27,37,33,43)(28,48,34,42)(29,47,35,41)(30,46,36,40)(49,86,55,92)(50,85,56,91)(51,96,57,90)(52,95,58,89)(53,94,59,88)(54,93,60,87)(61,187,67,181)(62,186,68,192)(63,185,69,191)(64,184,70,190)(65,183,71,189)(66,182,72,188)(73,158,79,164)(74,157,80,163)(75,168,81,162)(76,167,82,161)(77,166,83,160)(78,165,84,159)(97,124,103,130)(98,123,104,129)(99,122,105,128)(100,121,106,127)(101,132,107,126)(102,131,108,125)(109,170,115,176)(110,169,116,175)(111,180,117,174)(112,179,118,173)(113,178,119,172)(114,177,120,171) );
G=PermutationGroup([[(1,162,150,172),(2,163,151,173),(3,164,152,174),(4,165,153,175),(5,166,154,176),(6,167,155,177),(7,168,156,178),(8,157,145,179),(9,158,146,180),(10,159,147,169),(11,160,148,170),(12,161,149,171),(13,75,142,113),(14,76,143,114),(15,77,144,115),(16,78,133,116),(17,79,134,117),(18,80,135,118),(19,81,136,119),(20,82,137,120),(21,83,138,109),(22,84,139,110),(23,73,140,111),(24,74,141,112),(25,57,132,63),(26,58,121,64),(27,59,122,65),(28,60,123,66),(29,49,124,67),(30,50,125,68),(31,51,126,69),(32,52,127,70),(33,53,128,71),(34,54,129,72),(35,55,130,61),(36,56,131,62),(37,183,105,88),(38,184,106,89),(39,185,107,90),(40,186,108,91),(41,187,97,92),(42,188,98,93),(43,189,99,94),(44,190,100,95),(45,191,101,96),(46,192,102,85),(47,181,103,86),(48,182,104,87)], [(1,186,150,91),(2,187,151,92),(3,188,152,93),(4,189,153,94),(5,190,154,95),(6,191,155,96),(7,192,156,85),(8,181,145,86),(9,182,146,87),(10,183,147,88),(11,184,148,89),(12,185,149,90),(13,30,142,125),(14,31,143,126),(15,32,144,127),(16,33,133,128),(17,34,134,129),(18,35,135,130),(19,36,136,131),(20,25,137,132),(21,26,138,121),(22,27,139,122),(23,28,140,123),(24,29,141,124),(37,169,105,159),(38,170,106,160),(39,171,107,161),(40,172,108,162),(41,173,97,163),(42,174,98,164),(43,175,99,165),(44,176,100,166),(45,177,101,167),(46,178,102,168),(47,179,103,157),(48,180,104,158),(49,74,67,112),(50,75,68,113),(51,76,69,114),(52,77,70,115),(53,78,71,116),(54,79,72,117),(55,80,61,118),(56,81,62,119),(57,82,63,120),(58,83,64,109),(59,84,65,110),(60,73,66,111)], [(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)], [(1,142,7,136),(2,141,8,135),(3,140,9,134),(4,139,10,133),(5,138,11,144),(6,137,12,143),(13,156,19,150),(14,155,20,149),(15,154,21,148),(16,153,22,147),(17,152,23,146),(18,151,24,145),(25,39,31,45),(26,38,32,44),(27,37,33,43),(28,48,34,42),(29,47,35,41),(30,46,36,40),(49,86,55,92),(50,85,56,91),(51,96,57,90),(52,95,58,89),(53,94,59,88),(54,93,60,87),(61,187,67,181),(62,186,68,192),(63,185,69,191),(64,184,70,190),(65,183,71,189),(66,182,72,188),(73,158,79,164),(74,157,80,163),(75,168,81,162),(76,167,82,161),(77,166,83,160),(78,165,84,159),(97,124,103,130),(98,123,104,129),(99,122,105,128),(100,121,106,127),(101,132,107,126),(102,131,108,125),(109,170,115,176),(110,169,116,175),(111,180,117,174),(112,179,118,173),(113,178,119,172),(114,177,120,171)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | 4K | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 24 | 24 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | - | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | D6 | D6 | D6 | SD16 | C4○D4 | C3⋊D4 | Dic6 | C4○D12 | C8.C22 | Q8⋊2S3 | Q8.14D6 |
| kernel | Q8⋊4Dic6 | C12⋊C8 | C12.Q8 | Q8⋊2Dic3 | C12⋊2Q8 | Q8×C12 | C4×Q8 | C2×C12 | C3×Q8 | C42 | C4⋊C4 | C2×Q8 | C12 | C12 | C2×C4 | Q8 | C4 | C6 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 4 | 4 | 4 | 1 | 2 | 2 |
Matrix representation of Q8⋊4Dic6 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 48 | 0 | 0 |
| 0 | 0 | 3 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 69 | 0 | 0 |
| 0 | 0 | 18 | 61 | 0 | 0 |
| 0 | 0 | 0 | 0 | 43 | 60 |
| 0 | 0 | 0 | 0 | 13 | 30 |
| 1 | 3 | 0 | 0 | 0 | 0 |
| 48 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 69 | 13 | 0 | 0 | 0 | 0 |
| 38 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 70 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 60 |
| 0 | 0 | 0 | 0 | 62 | 71 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,3,0,0,0,0,48,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,12,18,0,0,0,0,69,61,0,0,0,0,0,0,43,13,0,0,0,0,60,30],[1,48,0,0,0,0,3,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[69,38,0,0,0,0,13,4,0,0,0,0,0,0,72,70,0,0,0,0,0,1,0,0,0,0,0,0,2,62,0,0,0,0,60,71] >;
Q8⋊4Dic6 in GAP, Magma, Sage, TeX
Q_8\rtimes_4{\rm Dic}_6 % in TeX
G:=Group("Q8:4Dic6"); // GroupNames label
G:=SmallGroup(192,579);
// by ID
G=gap.SmallGroup(192,579);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,120,254,184,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^12=1,b^2=a^2,d^2=c^6,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^-1>;
// generators/relations