metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊2Dic6, Dic3.6SD16, (C3×Q8)⋊1Q8, C4⋊C4.15D6, C12⋊Q8.1C2, C12.5(C2×Q8), C3⋊1(Q8⋊Q8), (C2×C8).119D6, C8⋊Dic3.6C2, C4.5(C2×Dic6), Dic3⋊C8.5C2, (C2×Q8).123D6, Q8⋊C4.5S3, (Q8×Dic3).3C2, C6.27(C2×SD16), C2.14(S3×SD16), Q8⋊2Dic3.1C2, C2.8(Q16⋊S3), C22.185(S3×D4), C6.11(C22⋊Q8), C12.Q8.2C2, (C6×Q8).14C22, C12.157(C4○D4), C4.82(D4⋊2S3), (C2×C12).231C23, (C2×C24).130C22, (C2×Dic3).147D4, C6.53(C8.C22), C4⋊Dic3.81C22, (C4×Dic3).16C22, C2.16(Dic3.D4), (C2×C6).244(C2×D4), (C2×C3⋊C8).27C22, (C3×C4⋊C4).32C22, (C3×Q8⋊C4).5C2, (C2×C4).338(C22×S3), SmallGroup(192,350)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊2Dic6
G = < a,b,c,d | a4=c12=1, b2=a2, d2=c6, bab-1=cac-1=a-1, ad=da, cbc-1=ab, bd=db, dcd-1=c-1 >
Subgroups: 248 in 96 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Q8, Dic3, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×Q8, C2×Q8, C3⋊C8, C24, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Q8, Q8⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C4×Q8, C4⋊Q8, C2×C3⋊C8, C4×Dic3, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C3×C4⋊C4, C2×C24, C2×Dic6, C6×Q8, Q8⋊Q8, C12.Q8, Dic3⋊C8, C8⋊Dic3, Q8⋊2Dic3, C3×Q8⋊C4, C12⋊Q8, Q8×Dic3, Q8⋊2Dic6
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, SD16, C2×D4, C2×Q8, C4○D4, Dic6, C22×S3, C22⋊Q8, C2×SD16, C8.C22, C2×Dic6, S3×D4, D4⋊2S3, Q8⋊Q8, Dic3.D4, S3×SD16, Q16⋊S3, Q8⋊2Dic6
►Regular action on 192 points
Generators in S192
(1 166 89 21)(2 22 90 167)(3 168 91 23)(4 24 92 157)(5 158 93 13)(6 14 94 159)(7 160 95 15)(8 16 96 161)(9 162 85 17)(10 18 86 163)(11 164 87 19)(12 20 88 165)(25 175 46 156)(26 145 47 176)(27 177 48 146)(28 147 37 178)(29 179 38 148)(30 149 39 180)(31 169 40 150)(32 151 41 170)(33 171 42 152)(34 153 43 172)(35 173 44 154)(36 155 45 174)(49 65 98 142)(50 143 99 66)(51 67 100 144)(52 133 101 68)(53 69 102 134)(54 135 103 70)(55 71 104 136)(56 137 105 72)(57 61 106 138)(58 139 107 62)(59 63 108 140)(60 141 97 64)(73 131 119 191)(74 192 120 132)(75 121 109 181)(76 182 110 122)(77 123 111 183)(78 184 112 124)(79 125 113 185)(80 186 114 126)(81 127 115 187)(82 188 116 128)(83 129 117 189)(84 190 118 130)
(1 55 89 104)(2 137 90 72)(3 57 91 106)(4 139 92 62)(5 59 93 108)(6 141 94 64)(7 49 95 98)(8 143 96 66)(9 51 85 100)(10 133 86 68)(11 53 87 102)(12 135 88 70)(13 63 158 140)(14 60 159 97)(15 65 160 142)(16 50 161 99)(17 67 162 144)(18 52 163 101)(19 69 164 134)(20 54 165 103)(21 71 166 136)(22 56 167 105)(23 61 168 138)(24 58 157 107)(25 121 46 181)(26 76 47 110)(27 123 48 183)(28 78 37 112)(29 125 38 185)(30 80 39 114)(31 127 40 187)(32 82 41 116)(33 129 42 189)(34 84 43 118)(35 131 44 191)(36 74 45 120)(73 154 119 173)(75 156 109 175)(77 146 111 177)(79 148 113 179)(81 150 115 169)(83 152 117 171)(122 176 182 145)(124 178 184 147)(126 180 186 149)(128 170 188 151)(130 172 190 153)(132 174 192 155)
(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)
(1 118 7 112)(2 117 8 111)(3 116 9 110)(4 115 10 109)(5 114 11 120)(6 113 12 119)(13 186 19 192)(14 185 20 191)(15 184 21 190)(16 183 22 189)(17 182 23 188)(18 181 24 187)(25 58 31 52)(26 57 32 51)(27 56 33 50)(28 55 34 49)(29 54 35 60)(30 53 36 59)(37 104 43 98)(38 103 44 97)(39 102 45 108)(40 101 46 107)(41 100 47 106)(42 99 48 105)(61 151 67 145)(62 150 68 156)(63 149 69 155)(64 148 70 154)(65 147 71 153)(66 146 72 152)(73 94 79 88)(74 93 80 87)(75 92 81 86)(76 91 82 85)(77 90 83 96)(78 89 84 95)(121 157 127 163)(122 168 128 162)(123 167 129 161)(124 166 130 160)(125 165 131 159)(126 164 132 158)(133 175 139 169)(134 174 140 180)(135 173 141 179)(136 172 142 178)(137 171 143 177)(138 170 144 176)
G:=sub<Sym(192)| (1,166,89,21)(2,22,90,167)(3,168,91,23)(4,24,92,157)(5,158,93,13)(6,14,94,159)(7,160,95,15)(8,16,96,161)(9,162,85,17)(10,18,86,163)(11,164,87,19)(12,20,88,165)(25,175,46,156)(26,145,47,176)(27,177,48,146)(28,147,37,178)(29,179,38,148)(30,149,39,180)(31,169,40,150)(32,151,41,170)(33,171,42,152)(34,153,43,172)(35,173,44,154)(36,155,45,174)(49,65,98,142)(50,143,99,66)(51,67,100,144)(52,133,101,68)(53,69,102,134)(54,135,103,70)(55,71,104,136)(56,137,105,72)(57,61,106,138)(58,139,107,62)(59,63,108,140)(60,141,97,64)(73,131,119,191)(74,192,120,132)(75,121,109,181)(76,182,110,122)(77,123,111,183)(78,184,112,124)(79,125,113,185)(80,186,114,126)(81,127,115,187)(82,188,116,128)(83,129,117,189)(84,190,118,130), (1,55,89,104)(2,137,90,72)(3,57,91,106)(4,139,92,62)(5,59,93,108)(6,141,94,64)(7,49,95,98)(8,143,96,66)(9,51,85,100)(10,133,86,68)(11,53,87,102)(12,135,88,70)(13,63,158,140)(14,60,159,97)(15,65,160,142)(16,50,161,99)(17,67,162,144)(18,52,163,101)(19,69,164,134)(20,54,165,103)(21,71,166,136)(22,56,167,105)(23,61,168,138)(24,58,157,107)(25,121,46,181)(26,76,47,110)(27,123,48,183)(28,78,37,112)(29,125,38,185)(30,80,39,114)(31,127,40,187)(32,82,41,116)(33,129,42,189)(34,84,43,118)(35,131,44,191)(36,74,45,120)(73,154,119,173)(75,156,109,175)(77,146,111,177)(79,148,113,179)(81,150,115,169)(83,152,117,171)(122,176,182,145)(124,178,184,147)(126,180,186,149)(128,170,188,151)(130,172,190,153)(132,174,192,155), (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), (1,118,7,112)(2,117,8,111)(3,116,9,110)(4,115,10,109)(5,114,11,120)(6,113,12,119)(13,186,19,192)(14,185,20,191)(15,184,21,190)(16,183,22,189)(17,182,23,188)(18,181,24,187)(25,58,31,52)(26,57,32,51)(27,56,33,50)(28,55,34,49)(29,54,35,60)(30,53,36,59)(37,104,43,98)(38,103,44,97)(39,102,45,108)(40,101,46,107)(41,100,47,106)(42,99,48,105)(61,151,67,145)(62,150,68,156)(63,149,69,155)(64,148,70,154)(65,147,71,153)(66,146,72,152)(73,94,79,88)(74,93,80,87)(75,92,81,86)(76,91,82,85)(77,90,83,96)(78,89,84,95)(121,157,127,163)(122,168,128,162)(123,167,129,161)(124,166,130,160)(125,165,131,159)(126,164,132,158)(133,175,139,169)(134,174,140,180)(135,173,141,179)(136,172,142,178)(137,171,143,177)(138,170,144,176)>;
G:=Group( (1,166,89,21)(2,22,90,167)(3,168,91,23)(4,24,92,157)(5,158,93,13)(6,14,94,159)(7,160,95,15)(8,16,96,161)(9,162,85,17)(10,18,86,163)(11,164,87,19)(12,20,88,165)(25,175,46,156)(26,145,47,176)(27,177,48,146)(28,147,37,178)(29,179,38,148)(30,149,39,180)(31,169,40,150)(32,151,41,170)(33,171,42,152)(34,153,43,172)(35,173,44,154)(36,155,45,174)(49,65,98,142)(50,143,99,66)(51,67,100,144)(52,133,101,68)(53,69,102,134)(54,135,103,70)(55,71,104,136)(56,137,105,72)(57,61,106,138)(58,139,107,62)(59,63,108,140)(60,141,97,64)(73,131,119,191)(74,192,120,132)(75,121,109,181)(76,182,110,122)(77,123,111,183)(78,184,112,124)(79,125,113,185)(80,186,114,126)(81,127,115,187)(82,188,116,128)(83,129,117,189)(84,190,118,130), (1,55,89,104)(2,137,90,72)(3,57,91,106)(4,139,92,62)(5,59,93,108)(6,141,94,64)(7,49,95,98)(8,143,96,66)(9,51,85,100)(10,133,86,68)(11,53,87,102)(12,135,88,70)(13,63,158,140)(14,60,159,97)(15,65,160,142)(16,50,161,99)(17,67,162,144)(18,52,163,101)(19,69,164,134)(20,54,165,103)(21,71,166,136)(22,56,167,105)(23,61,168,138)(24,58,157,107)(25,121,46,181)(26,76,47,110)(27,123,48,183)(28,78,37,112)(29,125,38,185)(30,80,39,114)(31,127,40,187)(32,82,41,116)(33,129,42,189)(34,84,43,118)(35,131,44,191)(36,74,45,120)(73,154,119,173)(75,156,109,175)(77,146,111,177)(79,148,113,179)(81,150,115,169)(83,152,117,171)(122,176,182,145)(124,178,184,147)(126,180,186,149)(128,170,188,151)(130,172,190,153)(132,174,192,155), (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), (1,118,7,112)(2,117,8,111)(3,116,9,110)(4,115,10,109)(5,114,11,120)(6,113,12,119)(13,186,19,192)(14,185,20,191)(15,184,21,190)(16,183,22,189)(17,182,23,188)(18,181,24,187)(25,58,31,52)(26,57,32,51)(27,56,33,50)(28,55,34,49)(29,54,35,60)(30,53,36,59)(37,104,43,98)(38,103,44,97)(39,102,45,108)(40,101,46,107)(41,100,47,106)(42,99,48,105)(61,151,67,145)(62,150,68,156)(63,149,69,155)(64,148,70,154)(65,147,71,153)(66,146,72,152)(73,94,79,88)(74,93,80,87)(75,92,81,86)(76,91,82,85)(77,90,83,96)(78,89,84,95)(121,157,127,163)(122,168,128,162)(123,167,129,161)(124,166,130,160)(125,165,131,159)(126,164,132,158)(133,175,139,169)(134,174,140,180)(135,173,141,179)(136,172,142,178)(137,171,143,177)(138,170,144,176) );
G=PermutationGroup([[(1,166,89,21),(2,22,90,167),(3,168,91,23),(4,24,92,157),(5,158,93,13),(6,14,94,159),(7,160,95,15),(8,16,96,161),(9,162,85,17),(10,18,86,163),(11,164,87,19),(12,20,88,165),(25,175,46,156),(26,145,47,176),(27,177,48,146),(28,147,37,178),(29,179,38,148),(30,149,39,180),(31,169,40,150),(32,151,41,170),(33,171,42,152),(34,153,43,172),(35,173,44,154),(36,155,45,174),(49,65,98,142),(50,143,99,66),(51,67,100,144),(52,133,101,68),(53,69,102,134),(54,135,103,70),(55,71,104,136),(56,137,105,72),(57,61,106,138),(58,139,107,62),(59,63,108,140),(60,141,97,64),(73,131,119,191),(74,192,120,132),(75,121,109,181),(76,182,110,122),(77,123,111,183),(78,184,112,124),(79,125,113,185),(80,186,114,126),(81,127,115,187),(82,188,116,128),(83,129,117,189),(84,190,118,130)], [(1,55,89,104),(2,137,90,72),(3,57,91,106),(4,139,92,62),(5,59,93,108),(6,141,94,64),(7,49,95,98),(8,143,96,66),(9,51,85,100),(10,133,86,68),(11,53,87,102),(12,135,88,70),(13,63,158,140),(14,60,159,97),(15,65,160,142),(16,50,161,99),(17,67,162,144),(18,52,163,101),(19,69,164,134),(20,54,165,103),(21,71,166,136),(22,56,167,105),(23,61,168,138),(24,58,157,107),(25,121,46,181),(26,76,47,110),(27,123,48,183),(28,78,37,112),(29,125,38,185),(30,80,39,114),(31,127,40,187),(32,82,41,116),(33,129,42,189),(34,84,43,118),(35,131,44,191),(36,74,45,120),(73,154,119,173),(75,156,109,175),(77,146,111,177),(79,148,113,179),(81,150,115,169),(83,152,117,171),(122,176,182,145),(124,178,184,147),(126,180,186,149),(128,170,188,151),(130,172,190,153),(132,174,192,155)], [(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)], [(1,118,7,112),(2,117,8,111),(3,116,9,110),(4,115,10,109),(5,114,11,120),(6,113,12,119),(13,186,19,192),(14,185,20,191),(15,184,21,190),(16,183,22,189),(17,182,23,188),(18,181,24,187),(25,58,31,52),(26,57,32,51),(27,56,33,50),(28,55,34,49),(29,54,35,60),(30,53,36,59),(37,104,43,98),(38,103,44,97),(39,102,45,108),(40,101,46,107),(41,100,47,106),(42,99,48,105),(61,151,67,145),(62,150,68,156),(63,149,69,155),(64,148,70,154),(65,147,71,153),(66,146,72,152),(73,94,79,88),(74,93,80,87),(75,92,81,86),(76,91,82,85),(77,90,83,96),(78,89,84,95),(121,157,127,163),(122,168,128,162),(123,167,129,161),(124,166,130,160),(125,165,131,159),(126,164,132,158),(133,175,139,169),(134,174,140,180),(135,173,141,179),(136,172,142,178),(137,171,143,177),(138,170,144,176)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 8 | 12 | 12 | 12 | 24 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | - | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | D6 | D6 | D6 | SD16 | C4○D4 | Dic6 | C8.C22 | D4⋊2S3 | S3×D4 | S3×SD16 | Q16⋊S3 |
| kernel | Q8⋊2Dic6 | C12.Q8 | Dic3⋊C8 | C8⋊Dic3 | Q8⋊2Dic3 | C3×Q8⋊C4 | C12⋊Q8 | Q8×Dic3 | Q8⋊C4 | C2×Dic3 | C3×Q8 | C4⋊C4 | C2×C8 | C2×Q8 | Dic3 | C12 | Q8 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of Q8⋊2Dic6 ►in GL4(𝔽73) generated by
| 1 | 48 | 0 | 0 |
| 3 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 46 | 18 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 38 | 0 | 0 |
| 25 | 0 | 0 | 0 |
| 0 | 0 | 66 | 66 |
| 0 | 0 | 7 | 59 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 63 | 22 |
| 0 | 0 | 12 | 10 |
G:=sub<GL(4,GF(73))| [1,3,0,0,48,72,0,0,0,0,1,0,0,0,0,1],[46,0,0,0,18,27,0,0,0,0,72,0,0,0,0,72],[0,25,0,0,38,0,0,0,0,0,66,7,0,0,66,59],[72,0,0,0,0,72,0,0,0,0,63,12,0,0,22,10] >;
Q8⋊2Dic6 in GAP, Magma, Sage, TeX
Q_8\rtimes_2{\rm Dic}_6 % in TeX
G:=Group("Q8:2Dic6"); // GroupNames label
G:=SmallGroup(192,350);
// by ID
G=gap.SmallGroup(192,350);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,232,254,219,58,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^12=1,b^2=a^2,d^2=c^6,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations