metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊7Dic6, C42.124D6, C6.652- 1+4, (C3×Q8)⋊7Q8, C4⋊C4.293D6, C3⋊3(Q8⋊3Q8), (C4×Q8).19S3, C12.46(C2×Q8), (C2×Q8).222D6, (Q8×C12).13C2, C4.19(C2×Dic6), C6.17(C22×Q8), (C2×C6).114C24, C2.22(Q8○D12), C12⋊2Q8.25C2, (Q8×Dic3).12C2, (C4×Dic6).23C2, C12.333(C4○D4), (C4×C12).166C22, (C2×C12).168C23, C4.49(Q8⋊3S3), C4⋊Dic3.43C22, C4.Dic6.11C2, (C6×Q8).214C22, C2.19(C22×Dic6), C22.139(S3×C23), (C4×Dic3).83C22, (C2×Dic3).52C23, Dic3⋊C4.115C22, (C2×Dic6).242C22, C6.110(C2×C4○D4), C2.10(C2×Q8⋊3S3), (C3×C4⋊C4).342C22, (C2×C4).733(C22×S3), SmallGroup(192,1129)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊7Dic6
G = < a,b,c,d | a4=c12=1, b2=a2, d2=c6, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=a2b, dcd-1=c-1 >
Subgroups: 392 in 200 conjugacy classes, 115 normal (18 characteristic)
C1, C2, C3, C4, C4, C22, C6, C2×C4, C2×C4, C2×C4, Q8, Q8, Dic3, C12, C12, C2×C6, C42, C42, C4⋊C4, C4⋊C4, C2×Q8, C2×Q8, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C4×Q8, C4×Q8, C42.C2, C4⋊Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C4×C12, C3×C4⋊C4, C2×Dic6, C6×Q8, Q8⋊3Q8, C4×Dic6, C12⋊2Q8, C4.Dic6, Q8×Dic3, Q8×C12, Q8⋊7Dic6
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C4○D4, C24, Dic6, C22×S3, C22×Q8, C2×C4○D4, 2- 1+4, C2×Dic6, Q8⋊3S3, S3×C23, Q8⋊3Q8, C22×Dic6, C2×Q8⋊3S3, Q8○D12, Q8⋊7Dic6
►Regular action on 192 points
Generators in S192
(1 102 96 173)(2 103 85 174)(3 104 86 175)(4 105 87 176)(5 106 88 177)(6 107 89 178)(7 108 90 179)(8 97 91 180)(9 98 92 169)(10 99 93 170)(11 100 94 171)(12 101 95 172)(13 187 82 56)(14 188 83 57)(15 189 84 58)(16 190 73 59)(17 191 74 60)(18 192 75 49)(19 181 76 50)(20 182 77 51)(21 183 78 52)(22 184 79 53)(23 185 80 54)(24 186 81 55)(25 115 144 147)(26 116 133 148)(27 117 134 149)(28 118 135 150)(29 119 136 151)(30 120 137 152)(31 109 138 153)(32 110 139 154)(33 111 140 155)(34 112 141 156)(35 113 142 145)(36 114 143 146)(37 164 71 127)(38 165 72 128)(39 166 61 129)(40 167 62 130)(41 168 63 131)(42 157 64 132)(43 158 65 121)(44 159 66 122)(45 160 67 123)(46 161 68 124)(47 162 69 125)(48 163 70 126)
(1 125 96 162)(2 126 85 163)(3 127 86 164)(4 128 87 165)(5 129 88 166)(6 130 89 167)(7 131 90 168)(8 132 91 157)(9 121 92 158)(10 122 93 159)(11 123 94 160)(12 124 95 161)(13 143 82 36)(14 144 83 25)(15 133 84 26)(16 134 73 27)(17 135 74 28)(18 136 75 29)(19 137 76 30)(20 138 77 31)(21 139 78 32)(22 140 79 33)(23 141 80 34)(24 142 81 35)(37 104 71 175)(38 105 72 176)(39 106 61 177)(40 107 62 178)(41 108 63 179)(42 97 64 180)(43 98 65 169)(44 99 66 170)(45 100 67 171)(46 101 68 172)(47 102 69 173)(48 103 70 174)(49 151 192 119)(50 152 181 120)(51 153 182 109)(52 154 183 110)(53 155 184 111)(54 156 185 112)(55 145 186 113)(56 146 187 114)(57 147 188 115)(58 148 189 116)(59 149 190 117)(60 150 191 118)
(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 7 134)(2 139 8 133)(3 138 9 144)(4 137 10 143)(5 136 11 142)(6 135 12 141)(13 128 19 122)(14 127 20 121)(15 126 21 132)(16 125 22 131)(17 124 23 130)(18 123 24 129)(25 86 31 92)(26 85 32 91)(27 96 33 90)(28 95 34 89)(29 94 35 88)(30 93 36 87)(37 182 43 188)(38 181 44 187)(39 192 45 186)(40 191 46 185)(41 190 47 184)(42 189 48 183)(49 67 55 61)(50 66 56 72)(51 65 57 71)(52 64 58 70)(53 63 59 69)(54 62 60 68)(73 162 79 168)(74 161 80 167)(75 160 81 166)(76 159 82 165)(77 158 83 164)(78 157 84 163)(97 148 103 154)(98 147 104 153)(99 146 105 152)(100 145 106 151)(101 156 107 150)(102 155 108 149)(109 169 115 175)(110 180 116 174)(111 179 117 173)(112 178 118 172)(113 177 119 171)(114 176 120 170)
G:=sub<Sym(192)| (1,102,96,173)(2,103,85,174)(3,104,86,175)(4,105,87,176)(5,106,88,177)(6,107,89,178)(7,108,90,179)(8,97,91,180)(9,98,92,169)(10,99,93,170)(11,100,94,171)(12,101,95,172)(13,187,82,56)(14,188,83,57)(15,189,84,58)(16,190,73,59)(17,191,74,60)(18,192,75,49)(19,181,76,50)(20,182,77,51)(21,183,78,52)(22,184,79,53)(23,185,80,54)(24,186,81,55)(25,115,144,147)(26,116,133,148)(27,117,134,149)(28,118,135,150)(29,119,136,151)(30,120,137,152)(31,109,138,153)(32,110,139,154)(33,111,140,155)(34,112,141,156)(35,113,142,145)(36,114,143,146)(37,164,71,127)(38,165,72,128)(39,166,61,129)(40,167,62,130)(41,168,63,131)(42,157,64,132)(43,158,65,121)(44,159,66,122)(45,160,67,123)(46,161,68,124)(47,162,69,125)(48,163,70,126), (1,125,96,162)(2,126,85,163)(3,127,86,164)(4,128,87,165)(5,129,88,166)(6,130,89,167)(7,131,90,168)(8,132,91,157)(9,121,92,158)(10,122,93,159)(11,123,94,160)(12,124,95,161)(13,143,82,36)(14,144,83,25)(15,133,84,26)(16,134,73,27)(17,135,74,28)(18,136,75,29)(19,137,76,30)(20,138,77,31)(21,139,78,32)(22,140,79,33)(23,141,80,34)(24,142,81,35)(37,104,71,175)(38,105,72,176)(39,106,61,177)(40,107,62,178)(41,108,63,179)(42,97,64,180)(43,98,65,169)(44,99,66,170)(45,100,67,171)(46,101,68,172)(47,102,69,173)(48,103,70,174)(49,151,192,119)(50,152,181,120)(51,153,182,109)(52,154,183,110)(53,155,184,111)(54,156,185,112)(55,145,186,113)(56,146,187,114)(57,147,188,115)(58,148,189,116)(59,149,190,117)(60,150,191,118), (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), (1,140,7,134)(2,139,8,133)(3,138,9,144)(4,137,10,143)(5,136,11,142)(6,135,12,141)(13,128,19,122)(14,127,20,121)(15,126,21,132)(16,125,22,131)(17,124,23,130)(18,123,24,129)(25,86,31,92)(26,85,32,91)(27,96,33,90)(28,95,34,89)(29,94,35,88)(30,93,36,87)(37,182,43,188)(38,181,44,187)(39,192,45,186)(40,191,46,185)(41,190,47,184)(42,189,48,183)(49,67,55,61)(50,66,56,72)(51,65,57,71)(52,64,58,70)(53,63,59,69)(54,62,60,68)(73,162,79,168)(74,161,80,167)(75,160,81,166)(76,159,82,165)(77,158,83,164)(78,157,84,163)(97,148,103,154)(98,147,104,153)(99,146,105,152)(100,145,106,151)(101,156,107,150)(102,155,108,149)(109,169,115,175)(110,180,116,174)(111,179,117,173)(112,178,118,172)(113,177,119,171)(114,176,120,170)>;
G:=Group( (1,102,96,173)(2,103,85,174)(3,104,86,175)(4,105,87,176)(5,106,88,177)(6,107,89,178)(7,108,90,179)(8,97,91,180)(9,98,92,169)(10,99,93,170)(11,100,94,171)(12,101,95,172)(13,187,82,56)(14,188,83,57)(15,189,84,58)(16,190,73,59)(17,191,74,60)(18,192,75,49)(19,181,76,50)(20,182,77,51)(21,183,78,52)(22,184,79,53)(23,185,80,54)(24,186,81,55)(25,115,144,147)(26,116,133,148)(27,117,134,149)(28,118,135,150)(29,119,136,151)(30,120,137,152)(31,109,138,153)(32,110,139,154)(33,111,140,155)(34,112,141,156)(35,113,142,145)(36,114,143,146)(37,164,71,127)(38,165,72,128)(39,166,61,129)(40,167,62,130)(41,168,63,131)(42,157,64,132)(43,158,65,121)(44,159,66,122)(45,160,67,123)(46,161,68,124)(47,162,69,125)(48,163,70,126), (1,125,96,162)(2,126,85,163)(3,127,86,164)(4,128,87,165)(5,129,88,166)(6,130,89,167)(7,131,90,168)(8,132,91,157)(9,121,92,158)(10,122,93,159)(11,123,94,160)(12,124,95,161)(13,143,82,36)(14,144,83,25)(15,133,84,26)(16,134,73,27)(17,135,74,28)(18,136,75,29)(19,137,76,30)(20,138,77,31)(21,139,78,32)(22,140,79,33)(23,141,80,34)(24,142,81,35)(37,104,71,175)(38,105,72,176)(39,106,61,177)(40,107,62,178)(41,108,63,179)(42,97,64,180)(43,98,65,169)(44,99,66,170)(45,100,67,171)(46,101,68,172)(47,102,69,173)(48,103,70,174)(49,151,192,119)(50,152,181,120)(51,153,182,109)(52,154,183,110)(53,155,184,111)(54,156,185,112)(55,145,186,113)(56,146,187,114)(57,147,188,115)(58,148,189,116)(59,149,190,117)(60,150,191,118), (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), (1,140,7,134)(2,139,8,133)(3,138,9,144)(4,137,10,143)(5,136,11,142)(6,135,12,141)(13,128,19,122)(14,127,20,121)(15,126,21,132)(16,125,22,131)(17,124,23,130)(18,123,24,129)(25,86,31,92)(26,85,32,91)(27,96,33,90)(28,95,34,89)(29,94,35,88)(30,93,36,87)(37,182,43,188)(38,181,44,187)(39,192,45,186)(40,191,46,185)(41,190,47,184)(42,189,48,183)(49,67,55,61)(50,66,56,72)(51,65,57,71)(52,64,58,70)(53,63,59,69)(54,62,60,68)(73,162,79,168)(74,161,80,167)(75,160,81,166)(76,159,82,165)(77,158,83,164)(78,157,84,163)(97,148,103,154)(98,147,104,153)(99,146,105,152)(100,145,106,151)(101,156,107,150)(102,155,108,149)(109,169,115,175)(110,180,116,174)(111,179,117,173)(112,178,118,172)(113,177,119,171)(114,176,120,170) );
G=PermutationGroup([[(1,102,96,173),(2,103,85,174),(3,104,86,175),(4,105,87,176),(5,106,88,177),(6,107,89,178),(7,108,90,179),(8,97,91,180),(9,98,92,169),(10,99,93,170),(11,100,94,171),(12,101,95,172),(13,187,82,56),(14,188,83,57),(15,189,84,58),(16,190,73,59),(17,191,74,60),(18,192,75,49),(19,181,76,50),(20,182,77,51),(21,183,78,52),(22,184,79,53),(23,185,80,54),(24,186,81,55),(25,115,144,147),(26,116,133,148),(27,117,134,149),(28,118,135,150),(29,119,136,151),(30,120,137,152),(31,109,138,153),(32,110,139,154),(33,111,140,155),(34,112,141,156),(35,113,142,145),(36,114,143,146),(37,164,71,127),(38,165,72,128),(39,166,61,129),(40,167,62,130),(41,168,63,131),(42,157,64,132),(43,158,65,121),(44,159,66,122),(45,160,67,123),(46,161,68,124),(47,162,69,125),(48,163,70,126)], [(1,125,96,162),(2,126,85,163),(3,127,86,164),(4,128,87,165),(5,129,88,166),(6,130,89,167),(7,131,90,168),(8,132,91,157),(9,121,92,158),(10,122,93,159),(11,123,94,160),(12,124,95,161),(13,143,82,36),(14,144,83,25),(15,133,84,26),(16,134,73,27),(17,135,74,28),(18,136,75,29),(19,137,76,30),(20,138,77,31),(21,139,78,32),(22,140,79,33),(23,141,80,34),(24,142,81,35),(37,104,71,175),(38,105,72,176),(39,106,61,177),(40,107,62,178),(41,108,63,179),(42,97,64,180),(43,98,65,169),(44,99,66,170),(45,100,67,171),(46,101,68,172),(47,102,69,173),(48,103,70,174),(49,151,192,119),(50,152,181,120),(51,153,182,109),(52,154,183,110),(53,155,184,111),(54,156,185,112),(55,145,186,113),(56,146,187,114),(57,147,188,115),(58,148,189,116),(59,149,190,117),(60,150,191,118)], [(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)], [(1,140,7,134),(2,139,8,133),(3,138,9,144),(4,137,10,143),(5,136,11,142),(6,135,12,141),(13,128,19,122),(14,127,20,121),(15,126,21,132),(16,125,22,131),(17,124,23,130),(18,123,24,129),(25,86,31,92),(26,85,32,91),(27,96,33,90),(28,95,34,89),(29,94,35,88),(30,93,36,87),(37,182,43,188),(38,181,44,187),(39,192,45,186),(40,191,46,185),(41,190,47,184),(42,189,48,183),(49,67,55,61),(50,66,56,72),(51,65,57,71),(52,64,58,70),(53,63,59,69),(54,62,60,68),(73,162,79,168),(74,161,80,167),(75,160,81,166),(76,159,82,165),(77,158,83,164),(78,157,84,163),(97,148,103,154),(98,147,104,153),(99,146,105,152),(100,145,106,151),(101,156,107,150),(102,155,108,149),(109,169,115,175),(110,180,116,174),(111,179,117,173),(112,178,118,172),(113,177,119,171),(114,176,120,170)]])
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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | - | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | D6 | C4○D4 | Dic6 | 2- 1+4 | Q8⋊3S3 | Q8○D12 |
| kernel | Q8⋊7Dic6 | C4×Dic6 | C12⋊2Q8 | C4.Dic6 | Q8×Dic3 | Q8×C12 | C4×Q8 | C3×Q8 | C42 | C4⋊C4 | C2×Q8 | C12 | Q8 | C6 | C4 | C2 |
| # reps | 1 | 3 | 3 | 6 | 2 | 1 | 1 | 4 | 3 | 3 | 1 | 4 | 8 | 1 | 2 | 2 |
Matrix representation of Q8⋊7Dic6 ►in GL4(𝔽13) generated by
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 12 | 12 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 6 |
| 0 | 0 | 4 | 1 |
| 6 | 3 | 0 | 0 |
| 10 | 3 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 11 | 2 | 0 | 0 |
| 4 | 2 | 0 | 0 |
| 0 | 0 | 5 | 10 |
| 0 | 0 | 8 | 8 |
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,1,12,0,0,2,12],[12,0,0,0,0,12,0,0,0,0,12,4,0,0,6,1],[6,10,0,0,3,3,0,0,0,0,12,0,0,0,0,12],[11,4,0,0,2,2,0,0,0,0,5,8,0,0,10,8] >;
Q8⋊7Dic6 in GAP, Magma, Sage, TeX
Q_8\rtimes_7{\rm Dic}_6 % in TeX
G:=Group("Q8:7Dic6"); // GroupNames label
G:=SmallGroup(192,1129);
// by ID
G=gap.SmallGroup(192,1129);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,232,387,184,1571,192,6278]);
// 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,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations