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), (C2×Q8).221D6, (Q8×C12).12C2, C4.18(C2×Dic6), C6.16(C22×Q8), (C2×C6).113C24, (C4×Dic6).22C2, (Q8×Dic3).11C2, (C2×C12).493C23, (C4×C12).165C22, Dic3⋊C4.9C22, C4.Dic6.10C2, C12.6Q8.11C2, (C6×Q8).213C22, C2.18(C22×Dic6), Dic3.36(C4○D4), C4⋊Dic3.304C22, C22.138(S3×C23), (C4×Dic3).82C22, (C2×Dic3).51C23, 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
Generators and relations for Q8⋊6Dic6
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 >
Subgroups: 392 in 200 conjugacy classes, 113 normal (22 characteristic)
C1, C2, C3, C4, C4, C22, C6, C2×C4, C2×C4, C2×C4, Q8, Q8, Dic3, Dic3, C12, C12, C2×C6, C42, C42, C4⋊C4, C4⋊C4, C2×Q8, C2×Q8, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C4×Q8, C4×Q8, C42.C2, C4⋊Q8, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, C4×C12, C3×C4⋊C4, C2×Dic6, C6×Q8, Q8⋊3Q8, C4×Dic6, C12.6Q8, C12⋊Q8, C4.Dic6, Q8×Dic3, Q8×C12, Q8⋊6Dic6
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, S3×C23, Q8⋊3Q8, C22×Dic6, Q8.15D6, S3×C4○D4, Q8⋊6Dic6
►Regular action on 192 points
Generators in S192
(1 80 156 126)(2 81 145 127)(3 82 146 128)(4 83 147 129)(5 84 148 130)(6 73 149 131)(7 74 150 132)(8 75 151 121)(9 76 152 122)(10 77 153 123)(11 78 154 124)(12 79 155 125)(13 137 34 92)(14 138 35 93)(15 139 36 94)(16 140 25 95)(17 141 26 96)(18 142 27 85)(19 143 28 86)(20 144 29 87)(21 133 30 88)(22 134 31 89)(23 135 32 90)(24 136 33 91)(37 105 168 115)(38 106 157 116)(39 107 158 117)(40 108 159 118)(41 97 160 119)(42 98 161 120)(43 99 162 109)(44 100 163 110)(45 101 164 111)(46 102 165 112)(47 103 166 113)(48 104 167 114)(49 191 179 67)(50 192 180 68)(51 181 169 69)(52 182 170 70)(53 183 171 71)(54 184 172 72)(55 185 173 61)(56 186 174 62)(57 187 175 63)(58 188 176 64)(59 189 177 65)(60 190 178 66)
(1 136 156 91)(2 92 145 137)(3 138 146 93)(4 94 147 139)(5 140 148 95)(6 96 149 141)(7 142 150 85)(8 86 151 143)(9 144 152 87)(10 88 153 133)(11 134 154 89)(12 90 155 135)(13 81 34 127)(14 128 35 82)(15 83 36 129)(16 130 25 84)(17 73 26 131)(18 132 27 74)(19 75 28 121)(20 122 29 76)(21 77 30 123)(22 124 31 78)(23 79 32 125)(24 126 33 80)(37 190 168 66)(38 67 157 191)(39 192 158 68)(40 69 159 181)(41 182 160 70)(42 71 161 183)(43 184 162 72)(44 61 163 185)(45 186 164 62)(46 63 165 187)(47 188 166 64)(48 65 167 189)(49 106 179 116)(50 117 180 107)(51 108 169 118)(52 119 170 97)(53 98 171 120)(54 109 172 99)(55 100 173 110)(56 111 174 101)(57 102 175 112)(58 113 176 103)(59 104 177 114)(60 115 178 105)
(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)
(1 113 7 119)(2 112 8 118)(3 111 9 117)(4 110 10 116)(5 109 11 115)(6 120 12 114)(13 187 19 181)(14 186 20 192)(15 185 21 191)(16 184 22 190)(17 183 23 189)(18 182 24 188)(25 72 31 66)(26 71 32 65)(27 70 33 64)(28 69 34 63)(29 68 35 62)(30 67 36 61)(37 84 43 78)(38 83 44 77)(39 82 45 76)(40 81 46 75)(41 80 47 74)(42 79 48 73)(49 94 55 88)(50 93 56 87)(51 92 57 86)(52 91 58 85)(53 90 59 96)(54 89 60 95)(97 156 103 150)(98 155 104 149)(99 154 105 148)(100 153 106 147)(101 152 107 146)(102 151 108 145)(121 159 127 165)(122 158 128 164)(123 157 129 163)(124 168 130 162)(125 167 131 161)(126 166 132 160)(133 179 139 173)(134 178 140 172)(135 177 141 171)(136 176 142 170)(137 175 143 169)(138 174 144 180)
G:=sub<Sym(192)| (1,80,156,126)(2,81,145,127)(3,82,146,128)(4,83,147,129)(5,84,148,130)(6,73,149,131)(7,74,150,132)(8,75,151,121)(9,76,152,122)(10,77,153,123)(11,78,154,124)(12,79,155,125)(13,137,34,92)(14,138,35,93)(15,139,36,94)(16,140,25,95)(17,141,26,96)(18,142,27,85)(19,143,28,86)(20,144,29,87)(21,133,30,88)(22,134,31,89)(23,135,32,90)(24,136,33,91)(37,105,168,115)(38,106,157,116)(39,107,158,117)(40,108,159,118)(41,97,160,119)(42,98,161,120)(43,99,162,109)(44,100,163,110)(45,101,164,111)(46,102,165,112)(47,103,166,113)(48,104,167,114)(49,191,179,67)(50,192,180,68)(51,181,169,69)(52,182,170,70)(53,183,171,71)(54,184,172,72)(55,185,173,61)(56,186,174,62)(57,187,175,63)(58,188,176,64)(59,189,177,65)(60,190,178,66), (1,136,156,91)(2,92,145,137)(3,138,146,93)(4,94,147,139)(5,140,148,95)(6,96,149,141)(7,142,150,85)(8,86,151,143)(9,144,152,87)(10,88,153,133)(11,134,154,89)(12,90,155,135)(13,81,34,127)(14,128,35,82)(15,83,36,129)(16,130,25,84)(17,73,26,131)(18,132,27,74)(19,75,28,121)(20,122,29,76)(21,77,30,123)(22,124,31,78)(23,79,32,125)(24,126,33,80)(37,190,168,66)(38,67,157,191)(39,192,158,68)(40,69,159,181)(41,182,160,70)(42,71,161,183)(43,184,162,72)(44,61,163,185)(45,186,164,62)(46,63,165,187)(47,188,166,64)(48,65,167,189)(49,106,179,116)(50,117,180,107)(51,108,169,118)(52,119,170,97)(53,98,171,120)(54,109,172,99)(55,100,173,110)(56,111,174,101)(57,102,175,112)(58,113,176,103)(59,104,177,114)(60,115,178,105), (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), (1,113,7,119)(2,112,8,118)(3,111,9,117)(4,110,10,116)(5,109,11,115)(6,120,12,114)(13,187,19,181)(14,186,20,192)(15,185,21,191)(16,184,22,190)(17,183,23,189)(18,182,24,188)(25,72,31,66)(26,71,32,65)(27,70,33,64)(28,69,34,63)(29,68,35,62)(30,67,36,61)(37,84,43,78)(38,83,44,77)(39,82,45,76)(40,81,46,75)(41,80,47,74)(42,79,48,73)(49,94,55,88)(50,93,56,87)(51,92,57,86)(52,91,58,85)(53,90,59,96)(54,89,60,95)(97,156,103,150)(98,155,104,149)(99,154,105,148)(100,153,106,147)(101,152,107,146)(102,151,108,145)(121,159,127,165)(122,158,128,164)(123,157,129,163)(124,168,130,162)(125,167,131,161)(126,166,132,160)(133,179,139,173)(134,178,140,172)(135,177,141,171)(136,176,142,170)(137,175,143,169)(138,174,144,180)>;
G:=Group( (1,80,156,126)(2,81,145,127)(3,82,146,128)(4,83,147,129)(5,84,148,130)(6,73,149,131)(7,74,150,132)(8,75,151,121)(9,76,152,122)(10,77,153,123)(11,78,154,124)(12,79,155,125)(13,137,34,92)(14,138,35,93)(15,139,36,94)(16,140,25,95)(17,141,26,96)(18,142,27,85)(19,143,28,86)(20,144,29,87)(21,133,30,88)(22,134,31,89)(23,135,32,90)(24,136,33,91)(37,105,168,115)(38,106,157,116)(39,107,158,117)(40,108,159,118)(41,97,160,119)(42,98,161,120)(43,99,162,109)(44,100,163,110)(45,101,164,111)(46,102,165,112)(47,103,166,113)(48,104,167,114)(49,191,179,67)(50,192,180,68)(51,181,169,69)(52,182,170,70)(53,183,171,71)(54,184,172,72)(55,185,173,61)(56,186,174,62)(57,187,175,63)(58,188,176,64)(59,189,177,65)(60,190,178,66), (1,136,156,91)(2,92,145,137)(3,138,146,93)(4,94,147,139)(5,140,148,95)(6,96,149,141)(7,142,150,85)(8,86,151,143)(9,144,152,87)(10,88,153,133)(11,134,154,89)(12,90,155,135)(13,81,34,127)(14,128,35,82)(15,83,36,129)(16,130,25,84)(17,73,26,131)(18,132,27,74)(19,75,28,121)(20,122,29,76)(21,77,30,123)(22,124,31,78)(23,79,32,125)(24,126,33,80)(37,190,168,66)(38,67,157,191)(39,192,158,68)(40,69,159,181)(41,182,160,70)(42,71,161,183)(43,184,162,72)(44,61,163,185)(45,186,164,62)(46,63,165,187)(47,188,166,64)(48,65,167,189)(49,106,179,116)(50,117,180,107)(51,108,169,118)(52,119,170,97)(53,98,171,120)(54,109,172,99)(55,100,173,110)(56,111,174,101)(57,102,175,112)(58,113,176,103)(59,104,177,114)(60,115,178,105), (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), (1,113,7,119)(2,112,8,118)(3,111,9,117)(4,110,10,116)(5,109,11,115)(6,120,12,114)(13,187,19,181)(14,186,20,192)(15,185,21,191)(16,184,22,190)(17,183,23,189)(18,182,24,188)(25,72,31,66)(26,71,32,65)(27,70,33,64)(28,69,34,63)(29,68,35,62)(30,67,36,61)(37,84,43,78)(38,83,44,77)(39,82,45,76)(40,81,46,75)(41,80,47,74)(42,79,48,73)(49,94,55,88)(50,93,56,87)(51,92,57,86)(52,91,58,85)(53,90,59,96)(54,89,60,95)(97,156,103,150)(98,155,104,149)(99,154,105,148)(100,153,106,147)(101,152,107,146)(102,151,108,145)(121,159,127,165)(122,158,128,164)(123,157,129,163)(124,168,130,162)(125,167,131,161)(126,166,132,160)(133,179,139,173)(134,178,140,172)(135,177,141,171)(136,176,142,170)(137,175,143,169)(138,174,144,180) );
G=PermutationGroup([[(1,80,156,126),(2,81,145,127),(3,82,146,128),(4,83,147,129),(5,84,148,130),(6,73,149,131),(7,74,150,132),(8,75,151,121),(9,76,152,122),(10,77,153,123),(11,78,154,124),(12,79,155,125),(13,137,34,92),(14,138,35,93),(15,139,36,94),(16,140,25,95),(17,141,26,96),(18,142,27,85),(19,143,28,86),(20,144,29,87),(21,133,30,88),(22,134,31,89),(23,135,32,90),(24,136,33,91),(37,105,168,115),(38,106,157,116),(39,107,158,117),(40,108,159,118),(41,97,160,119),(42,98,161,120),(43,99,162,109),(44,100,163,110),(45,101,164,111),(46,102,165,112),(47,103,166,113),(48,104,167,114),(49,191,179,67),(50,192,180,68),(51,181,169,69),(52,182,170,70),(53,183,171,71),(54,184,172,72),(55,185,173,61),(56,186,174,62),(57,187,175,63),(58,188,176,64),(59,189,177,65),(60,190,178,66)], [(1,136,156,91),(2,92,145,137),(3,138,146,93),(4,94,147,139),(5,140,148,95),(6,96,149,141),(7,142,150,85),(8,86,151,143),(9,144,152,87),(10,88,153,133),(11,134,154,89),(12,90,155,135),(13,81,34,127),(14,128,35,82),(15,83,36,129),(16,130,25,84),(17,73,26,131),(18,132,27,74),(19,75,28,121),(20,122,29,76),(21,77,30,123),(22,124,31,78),(23,79,32,125),(24,126,33,80),(37,190,168,66),(38,67,157,191),(39,192,158,68),(40,69,159,181),(41,182,160,70),(42,71,161,183),(43,184,162,72),(44,61,163,185),(45,186,164,62),(46,63,165,187),(47,188,166,64),(48,65,167,189),(49,106,179,116),(50,117,180,107),(51,108,169,118),(52,119,170,97),(53,98,171,120),(54,109,172,99),(55,100,173,110),(56,111,174,101),(57,102,175,112),(58,113,176,103),(59,104,177,114),(60,115,178,105)], [(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)], [(1,113,7,119),(2,112,8,118),(3,111,9,117),(4,110,10,116),(5,109,11,115),(6,120,12,114),(13,187,19,181),(14,186,20,192),(15,185,21,191),(16,184,22,190),(17,183,23,189),(18,182,24,188),(25,72,31,66),(26,71,32,65),(27,70,33,64),(28,69,34,63),(29,68,35,62),(30,67,36,61),(37,84,43,78),(38,83,44,77),(39,82,45,76),(40,81,46,75),(41,80,47,74),(42,79,48,73),(49,94,55,88),(50,93,56,87),(51,92,57,86),(52,91,58,85),(53,90,59,96),(54,89,60,95),(97,156,103,150),(98,155,104,149),(99,154,105,148),(100,153,106,147),(101,152,107,146),(102,151,108,145),(121,159,127,165),(122,158,128,164),(123,157,129,163),(124,168,130,162),(125,167,131,161),(126,166,132,160),(133,179,139,173),(134,178,140,172),(135,177,141,171),(136,176,142,170),(137,175,143,169),(138,174,144,180)]])
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 | C4.Dic6 | 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 |
Matrix representation of Q8⋊6Dic6 ►in 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] >;
Q8⋊6Dic6 in GAP, Magma, Sage, TeX
Q_8\rtimes_6{\rm Dic}_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