metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6⋊8Q8, C42.169D6, C6.332- 1+4, C4⋊Q8.14S3, C4.17(S3×Q8), C4⋊C4.121D6, C3⋊5(Q8⋊3Q8), C12.52(C2×Q8), (C2×Q8).165D6, C6.44(C22×Q8), (C2×C6).265C24, (C2×C12).98C23, (Q8×Dic3).13C2, (C4×Dic6).25C2, Dic3.14(C2×Q8), Dic3.Q8.4C2, C12.135(C4○D4), C4.18(D4⋊2S3), (C4×C12).206C22, Dic3⋊Q8.8C2, C4.Dic6.14C2, (C6×Q8).132C22, Dic6⋊C4.12C2, Dic3⋊C4.57C22, C4⋊Dic3.382C22, C22.286(S3×C23), C2.34(Q8.15D6), (C2×Dic3).270C23, (C4×Dic3).157C22, (C2×Dic6).302C22, C2.27(C2×S3×Q8), C6.99(C2×C4○D4), (C3×C4⋊Q8).14C2, C2.63(C2×D4⋊2S3), (C2×C4).90(C22×S3), (C3×C4⋊C4).208C22, SmallGroup(192,1280)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic6⋊8Q8
G = < a,b,c,d | a12=c4=1, b2=a6, d2=c2, bab-1=a-1, ac=ca, dad-1=a7, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 384 in 200 conjugacy classes, 107 normal (27 characteristic)
C1, C2, C3, C4, C4, C22, C6, C2×C4, C2×C4, C2×C4, Q8, Dic3, Dic3, C12, C12, C2×C6, C42, C42, C4⋊C4, C4⋊C4, C2×Q8, C2×Q8, Dic6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C4×Q8, C42.C2, C4⋊Q8, C4⋊Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C4×C12, C3×C4⋊C4, C2×Dic6, C6×Q8, Q8⋊3Q8, C4×Dic6, Dic6⋊C4, Dic3.Q8, C4.Dic6, Dic3⋊Q8, Q8×Dic3, C3×C4⋊Q8, Dic6⋊8Q8
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C4○D4, C24, C22×S3, C22×Q8, C2×C4○D4, 2- 1+4, D4⋊2S3, S3×Q8, S3×C23, Q8⋊3Q8, C2×D4⋊2S3, C2×S3×Q8, Q8.15D6, Dic6⋊8Q8
►Regular action on 192 points
Generators in S192
(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 82 7 76)(2 81 8 75)(3 80 9 74)(4 79 10 73)(5 78 11 84)(6 77 12 83)(13 85 19 91)(14 96 20 90)(15 95 21 89)(16 94 22 88)(17 93 23 87)(18 92 24 86)(25 97 31 103)(26 108 32 102)(27 107 33 101)(28 106 34 100)(29 105 35 99)(30 104 36 98)(37 132 43 126)(38 131 44 125)(39 130 45 124)(40 129 46 123)(41 128 47 122)(42 127 48 121)(49 146 55 152)(50 145 56 151)(51 156 57 150)(52 155 58 149)(53 154 59 148)(54 153 60 147)(61 169 67 175)(62 180 68 174)(63 179 69 173)(64 178 70 172)(65 177 71 171)(66 176 72 170)(109 140 115 134)(110 139 116 133)(111 138 117 144)(112 137 118 143)(113 136 119 142)(114 135 120 141)(157 188 163 182)(158 187 164 181)(159 186 165 192)(160 185 166 191)(161 184 167 190)(162 183 168 189)
(1 102 172 115)(2 103 173 116)(3 104 174 117)(4 105 175 118)(5 106 176 119)(6 107 177 120)(7 108 178 109)(8 97 179 110)(9 98 180 111)(10 99 169 112)(11 100 170 113)(12 101 171 114)(13 39 51 163)(14 40 52 164)(15 41 53 165)(16 42 54 166)(17 43 55 167)(18 44 56 168)(19 45 57 157)(20 46 58 158)(21 47 59 159)(22 48 60 160)(23 37 49 161)(24 38 50 162)(25 63 133 81)(26 64 134 82)(27 65 135 83)(28 66 136 84)(29 67 137 73)(30 68 138 74)(31 69 139 75)(32 70 140 76)(33 71 141 77)(34 72 142 78)(35 61 143 79)(36 62 144 80)(85 130 156 182)(86 131 145 183)(87 132 146 184)(88 121 147 185)(89 122 148 186)(90 123 149 187)(91 124 150 188)(92 125 151 189)(93 126 152 190)(94 127 153 191)(95 128 154 192)(96 129 155 181)
(1 188 172 124)(2 183 173 131)(3 190 174 126)(4 185 175 121)(5 192 176 128)(6 187 177 123)(7 182 178 130)(8 189 179 125)(9 184 180 132)(10 191 169 127)(11 186 170 122)(12 181 171 129)(13 26 51 134)(14 33 52 141)(15 28 53 136)(16 35 54 143)(17 30 55 138)(18 25 56 133)(19 32 57 140)(20 27 58 135)(21 34 59 142)(22 29 60 137)(23 36 49 144)(24 31 50 139)(37 80 161 62)(38 75 162 69)(39 82 163 64)(40 77 164 71)(41 84 165 66)(42 79 166 61)(43 74 167 68)(44 81 168 63)(45 76 157 70)(46 83 158 65)(47 78 159 72)(48 73 160 67)(85 108 156 109)(86 103 145 116)(87 98 146 111)(88 105 147 118)(89 100 148 113)(90 107 149 120)(91 102 150 115)(92 97 151 110)(93 104 152 117)(94 99 153 112)(95 106 154 119)(96 101 155 114)
G:=sub<Sym(192)| (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), (1,82,7,76)(2,81,8,75)(3,80,9,74)(4,79,10,73)(5,78,11,84)(6,77,12,83)(13,85,19,91)(14,96,20,90)(15,95,21,89)(16,94,22,88)(17,93,23,87)(18,92,24,86)(25,97,31,103)(26,108,32,102)(27,107,33,101)(28,106,34,100)(29,105,35,99)(30,104,36,98)(37,132,43,126)(38,131,44,125)(39,130,45,124)(40,129,46,123)(41,128,47,122)(42,127,48,121)(49,146,55,152)(50,145,56,151)(51,156,57,150)(52,155,58,149)(53,154,59,148)(54,153,60,147)(61,169,67,175)(62,180,68,174)(63,179,69,173)(64,178,70,172)(65,177,71,171)(66,176,72,170)(109,140,115,134)(110,139,116,133)(111,138,117,144)(112,137,118,143)(113,136,119,142)(114,135,120,141)(157,188,163,182)(158,187,164,181)(159,186,165,192)(160,185,166,191)(161,184,167,190)(162,183,168,189), (1,102,172,115)(2,103,173,116)(3,104,174,117)(4,105,175,118)(5,106,176,119)(6,107,177,120)(7,108,178,109)(8,97,179,110)(9,98,180,111)(10,99,169,112)(11,100,170,113)(12,101,171,114)(13,39,51,163)(14,40,52,164)(15,41,53,165)(16,42,54,166)(17,43,55,167)(18,44,56,168)(19,45,57,157)(20,46,58,158)(21,47,59,159)(22,48,60,160)(23,37,49,161)(24,38,50,162)(25,63,133,81)(26,64,134,82)(27,65,135,83)(28,66,136,84)(29,67,137,73)(30,68,138,74)(31,69,139,75)(32,70,140,76)(33,71,141,77)(34,72,142,78)(35,61,143,79)(36,62,144,80)(85,130,156,182)(86,131,145,183)(87,132,146,184)(88,121,147,185)(89,122,148,186)(90,123,149,187)(91,124,150,188)(92,125,151,189)(93,126,152,190)(94,127,153,191)(95,128,154,192)(96,129,155,181), (1,188,172,124)(2,183,173,131)(3,190,174,126)(4,185,175,121)(5,192,176,128)(6,187,177,123)(7,182,178,130)(8,189,179,125)(9,184,180,132)(10,191,169,127)(11,186,170,122)(12,181,171,129)(13,26,51,134)(14,33,52,141)(15,28,53,136)(16,35,54,143)(17,30,55,138)(18,25,56,133)(19,32,57,140)(20,27,58,135)(21,34,59,142)(22,29,60,137)(23,36,49,144)(24,31,50,139)(37,80,161,62)(38,75,162,69)(39,82,163,64)(40,77,164,71)(41,84,165,66)(42,79,166,61)(43,74,167,68)(44,81,168,63)(45,76,157,70)(46,83,158,65)(47,78,159,72)(48,73,160,67)(85,108,156,109)(86,103,145,116)(87,98,146,111)(88,105,147,118)(89,100,148,113)(90,107,149,120)(91,102,150,115)(92,97,151,110)(93,104,152,117)(94,99,153,112)(95,106,154,119)(96,101,155,114)>;
G:=Group( (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), (1,82,7,76)(2,81,8,75)(3,80,9,74)(4,79,10,73)(5,78,11,84)(6,77,12,83)(13,85,19,91)(14,96,20,90)(15,95,21,89)(16,94,22,88)(17,93,23,87)(18,92,24,86)(25,97,31,103)(26,108,32,102)(27,107,33,101)(28,106,34,100)(29,105,35,99)(30,104,36,98)(37,132,43,126)(38,131,44,125)(39,130,45,124)(40,129,46,123)(41,128,47,122)(42,127,48,121)(49,146,55,152)(50,145,56,151)(51,156,57,150)(52,155,58,149)(53,154,59,148)(54,153,60,147)(61,169,67,175)(62,180,68,174)(63,179,69,173)(64,178,70,172)(65,177,71,171)(66,176,72,170)(109,140,115,134)(110,139,116,133)(111,138,117,144)(112,137,118,143)(113,136,119,142)(114,135,120,141)(157,188,163,182)(158,187,164,181)(159,186,165,192)(160,185,166,191)(161,184,167,190)(162,183,168,189), (1,102,172,115)(2,103,173,116)(3,104,174,117)(4,105,175,118)(5,106,176,119)(6,107,177,120)(7,108,178,109)(8,97,179,110)(9,98,180,111)(10,99,169,112)(11,100,170,113)(12,101,171,114)(13,39,51,163)(14,40,52,164)(15,41,53,165)(16,42,54,166)(17,43,55,167)(18,44,56,168)(19,45,57,157)(20,46,58,158)(21,47,59,159)(22,48,60,160)(23,37,49,161)(24,38,50,162)(25,63,133,81)(26,64,134,82)(27,65,135,83)(28,66,136,84)(29,67,137,73)(30,68,138,74)(31,69,139,75)(32,70,140,76)(33,71,141,77)(34,72,142,78)(35,61,143,79)(36,62,144,80)(85,130,156,182)(86,131,145,183)(87,132,146,184)(88,121,147,185)(89,122,148,186)(90,123,149,187)(91,124,150,188)(92,125,151,189)(93,126,152,190)(94,127,153,191)(95,128,154,192)(96,129,155,181), (1,188,172,124)(2,183,173,131)(3,190,174,126)(4,185,175,121)(5,192,176,128)(6,187,177,123)(7,182,178,130)(8,189,179,125)(9,184,180,132)(10,191,169,127)(11,186,170,122)(12,181,171,129)(13,26,51,134)(14,33,52,141)(15,28,53,136)(16,35,54,143)(17,30,55,138)(18,25,56,133)(19,32,57,140)(20,27,58,135)(21,34,59,142)(22,29,60,137)(23,36,49,144)(24,31,50,139)(37,80,161,62)(38,75,162,69)(39,82,163,64)(40,77,164,71)(41,84,165,66)(42,79,166,61)(43,74,167,68)(44,81,168,63)(45,76,157,70)(46,83,158,65)(47,78,159,72)(48,73,160,67)(85,108,156,109)(86,103,145,116)(87,98,146,111)(88,105,147,118)(89,100,148,113)(90,107,149,120)(91,102,150,115)(92,97,151,110)(93,104,152,117)(94,99,153,112)(95,106,154,119)(96,101,155,114) );
G=PermutationGroup([[(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)], [(1,82,7,76),(2,81,8,75),(3,80,9,74),(4,79,10,73),(5,78,11,84),(6,77,12,83),(13,85,19,91),(14,96,20,90),(15,95,21,89),(16,94,22,88),(17,93,23,87),(18,92,24,86),(25,97,31,103),(26,108,32,102),(27,107,33,101),(28,106,34,100),(29,105,35,99),(30,104,36,98),(37,132,43,126),(38,131,44,125),(39,130,45,124),(40,129,46,123),(41,128,47,122),(42,127,48,121),(49,146,55,152),(50,145,56,151),(51,156,57,150),(52,155,58,149),(53,154,59,148),(54,153,60,147),(61,169,67,175),(62,180,68,174),(63,179,69,173),(64,178,70,172),(65,177,71,171),(66,176,72,170),(109,140,115,134),(110,139,116,133),(111,138,117,144),(112,137,118,143),(113,136,119,142),(114,135,120,141),(157,188,163,182),(158,187,164,181),(159,186,165,192),(160,185,166,191),(161,184,167,190),(162,183,168,189)], [(1,102,172,115),(2,103,173,116),(3,104,174,117),(4,105,175,118),(5,106,176,119),(6,107,177,120),(7,108,178,109),(8,97,179,110),(9,98,180,111),(10,99,169,112),(11,100,170,113),(12,101,171,114),(13,39,51,163),(14,40,52,164),(15,41,53,165),(16,42,54,166),(17,43,55,167),(18,44,56,168),(19,45,57,157),(20,46,58,158),(21,47,59,159),(22,48,60,160),(23,37,49,161),(24,38,50,162),(25,63,133,81),(26,64,134,82),(27,65,135,83),(28,66,136,84),(29,67,137,73),(30,68,138,74),(31,69,139,75),(32,70,140,76),(33,71,141,77),(34,72,142,78),(35,61,143,79),(36,62,144,80),(85,130,156,182),(86,131,145,183),(87,132,146,184),(88,121,147,185),(89,122,148,186),(90,123,149,187),(91,124,150,188),(92,125,151,189),(93,126,152,190),(94,127,153,191),(95,128,154,192),(96,129,155,181)], [(1,188,172,124),(2,183,173,131),(3,190,174,126),(4,185,175,121),(5,192,176,128),(6,187,177,123),(7,182,178,130),(8,189,179,125),(9,184,180,132),(10,191,169,127),(11,186,170,122),(12,181,171,129),(13,26,51,134),(14,33,52,141),(15,28,53,136),(16,35,54,143),(17,30,55,138),(18,25,56,133),(19,32,57,140),(20,27,58,135),(21,34,59,142),(22,29,60,137),(23,36,49,144),(24,31,50,139),(37,80,161,62),(38,75,162,69),(39,82,163,64),(40,77,164,71),(41,84,165,66),(42,79,166,61),(43,74,167,68),(44,81,168,63),(45,76,157,70),(46,83,158,65),(47,78,159,72),(48,73,160,67),(85,108,156,109),(86,103,145,116),(87,98,146,111),(88,105,147,118),(89,100,148,113),(90,107,149,120),(91,102,150,115),(92,97,151,110),(93,104,152,117),(94,99,153,112),(95,106,154,119),(96,101,155,114)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | ··· | 4Q | 4R | 4S | 4T | 4U | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 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 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | - | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | D6 | C4○D4 | 2- 1+4 | D4⋊2S3 | S3×Q8 | Q8.15D6 |
| kernel | Dic6⋊8Q8 | C4×Dic6 | Dic6⋊C4 | Dic3.Q8 | C4.Dic6 | Dic3⋊Q8 | Q8×Dic3 | C3×C4⋊Q8 | C4⋊Q8 | Dic6 | C42 | C4⋊C4 | C2×Q8 | C12 | C6 | C4 | C4 | C2 |
| # reps | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 1 | 1 | 4 | 1 | 4 | 2 | 4 | 1 | 2 | 2 | 2 |
Matrix representation of Dic6⋊8Q8 ►in GL6(𝔽13)
| 1 | 11 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 5 | 0 | 0 | 0 | 0 |
| 10 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 8 | 1 | 0 | 0 | 0 | 0 |
| 2 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 0 | 8 | 0 |
G:=sub<GL(6,GF(13))| [1,1,0,0,0,0,11,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,10,0,0,0,0,5,12,0,0,0,0,0,0,12,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,0],[8,2,0,0,0,0,1,5,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,8,0,0,0,0,8,0] >;
Dic6⋊8Q8 in GAP, Magma, Sage, TeX
{\rm Dic}_6\rtimes_8Q_8 % in TeX
G:=Group("Dic6:8Q8"); // GroupNames label
G:=SmallGroup(192,1280);
// by ID
G=gap.SmallGroup(192,1280);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,120,219,268,1571,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=c^4=1,b^2=a^6,d^2=c^2,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a^7,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations