metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊3Dic6, Dic3.3Q16, (C3×Q8)⋊2Q8, C4⋊C4.17D6, (C2×C8).13D6, C12⋊Q8.3C2, C12.6(C2×Q8), C2.7(S3×Q16), C3⋊1(C4.Q16), C6.14(C2×Q16), C24⋊1C4.6C2, C4.6(C2×Dic6), Dic3⋊C8.2C2, Q8⋊C4.2S3, (C2×Q8).124D6, (Q8×Dic3).4C2, C6.Q16.2C2, C2.14(Q8⋊3D6), C6.59(C8⋊C22), (C2×C24).13C22, Q8⋊2Dic3.3C2, C22.187(S3×D4), C6.12(C22⋊Q8), C12.158(C4○D4), (C6×Q8).16C22, C4.83(D4⋊2S3), (C2×C12).233C23, (C2×Dic3).148D4, C4⋊Dic3.83C22, (C4×Dic3).17C22, C2.17(Dic3.D4), (C2×C6).246(C2×D4), (C2×C3⋊C8).28C22, (C3×C4⋊C4).34C22, (C3×Q8⋊C4).2C2, (C2×C4).340(C22×S3), SmallGroup(192,352)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊3Dic6
G = < a,b,c,d | a4=c12=1, b2=a2, d2=c6, bab-1=cac-1=a-1, ad=da, cbc-1=ab, dbd-1=a2b, 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, C2.D8, 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, C4.Q16, C6.Q16, Dic3⋊C8, C24⋊1C4, Q8⋊2Dic3, C3×Q8⋊C4, C12⋊Q8, Q8×Dic3, Q8⋊3Dic6
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, Q16, C2×D4, C2×Q8, C4○D4, Dic6, C22×S3, C22⋊Q8, C2×Q16, C8⋊C22, C2×Dic6, S3×D4, D4⋊2S3, C4.Q16, Dic3.D4, Q8⋊3D6, S3×Q16, Q8⋊3Dic6
►Regular action on 192 points
Generators in S192
(1 71 44 117)(2 118 45 72)(3 61 46 119)(4 120 47 62)(5 63 48 109)(6 110 37 64)(7 65 38 111)(8 112 39 66)(9 67 40 113)(10 114 41 68)(11 69 42 115)(12 116 43 70)(13 50 73 156)(14 145 74 51)(15 52 75 146)(16 147 76 53)(17 54 77 148)(18 149 78 55)(19 56 79 150)(20 151 80 57)(21 58 81 152)(22 153 82 59)(23 60 83 154)(24 155 84 49)(25 104 184 172)(26 173 185 105)(27 106 186 174)(28 175 187 107)(29 108 188 176)(30 177 189 97)(31 98 190 178)(32 179 191 99)(33 100 192 180)(34 169 181 101)(35 102 182 170)(36 171 183 103)(85 158 138 124)(86 125 139 159)(87 160 140 126)(88 127 141 161)(89 162 142 128)(90 129 143 163)(91 164 144 130)(92 131 133 165)(93 166 134 132)(94 121 135 167)(95 168 136 122)(96 123 137 157)
(1 161 44 127)(2 142 45 89)(3 163 46 129)(4 144 47 91)(5 165 48 131)(6 134 37 93)(7 167 38 121)(8 136 39 95)(9 157 40 123)(10 138 41 85)(11 159 42 125)(12 140 43 87)(13 169 73 101)(14 35 74 182)(15 171 75 103)(16 25 76 184)(17 173 77 105)(18 27 78 186)(19 175 79 107)(20 29 80 188)(21 177 81 97)(22 31 82 190)(23 179 83 99)(24 33 84 192)(26 148 185 54)(28 150 187 56)(30 152 189 58)(32 154 191 60)(34 156 181 50)(36 146 183 52)(49 100 155 180)(51 102 145 170)(53 104 147 172)(55 106 149 174)(57 108 151 176)(59 98 153 178)(61 143 119 90)(62 130 120 164)(63 133 109 92)(64 132 110 166)(65 135 111 94)(66 122 112 168)(67 137 113 96)(68 124 114 158)(69 139 115 86)(70 126 116 160)(71 141 117 88)(72 128 118 162)
(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)
(1 185 7 191)(2 184 8 190)(3 183 9 189)(4 182 10 188)(5 181 11 187)(6 192 12 186)(13 86 19 92)(14 85 20 91)(15 96 21 90)(16 95 22 89)(17 94 23 88)(18 93 24 87)(25 39 31 45)(26 38 32 44)(27 37 33 43)(28 48 34 42)(29 47 35 41)(30 46 36 40)(49 126 55 132)(50 125 56 131)(51 124 57 130)(52 123 58 129)(53 122 59 128)(54 121 60 127)(61 103 67 97)(62 102 68 108)(63 101 69 107)(64 100 70 106)(65 99 71 105)(66 98 72 104)(73 139 79 133)(74 138 80 144)(75 137 81 143)(76 136 82 142)(77 135 83 141)(78 134 84 140)(109 169 115 175)(110 180 116 174)(111 179 117 173)(112 178 118 172)(113 177 119 171)(114 176 120 170)(145 158 151 164)(146 157 152 163)(147 168 153 162)(148 167 154 161)(149 166 155 160)(150 165 156 159)
G:=sub<Sym(192)| (1,71,44,117)(2,118,45,72)(3,61,46,119)(4,120,47,62)(5,63,48,109)(6,110,37,64)(7,65,38,111)(8,112,39,66)(9,67,40,113)(10,114,41,68)(11,69,42,115)(12,116,43,70)(13,50,73,156)(14,145,74,51)(15,52,75,146)(16,147,76,53)(17,54,77,148)(18,149,78,55)(19,56,79,150)(20,151,80,57)(21,58,81,152)(22,153,82,59)(23,60,83,154)(24,155,84,49)(25,104,184,172)(26,173,185,105)(27,106,186,174)(28,175,187,107)(29,108,188,176)(30,177,189,97)(31,98,190,178)(32,179,191,99)(33,100,192,180)(34,169,181,101)(35,102,182,170)(36,171,183,103)(85,158,138,124)(86,125,139,159)(87,160,140,126)(88,127,141,161)(89,162,142,128)(90,129,143,163)(91,164,144,130)(92,131,133,165)(93,166,134,132)(94,121,135,167)(95,168,136,122)(96,123,137,157), (1,161,44,127)(2,142,45,89)(3,163,46,129)(4,144,47,91)(5,165,48,131)(6,134,37,93)(7,167,38,121)(8,136,39,95)(9,157,40,123)(10,138,41,85)(11,159,42,125)(12,140,43,87)(13,169,73,101)(14,35,74,182)(15,171,75,103)(16,25,76,184)(17,173,77,105)(18,27,78,186)(19,175,79,107)(20,29,80,188)(21,177,81,97)(22,31,82,190)(23,179,83,99)(24,33,84,192)(26,148,185,54)(28,150,187,56)(30,152,189,58)(32,154,191,60)(34,156,181,50)(36,146,183,52)(49,100,155,180)(51,102,145,170)(53,104,147,172)(55,106,149,174)(57,108,151,176)(59,98,153,178)(61,143,119,90)(62,130,120,164)(63,133,109,92)(64,132,110,166)(65,135,111,94)(66,122,112,168)(67,137,113,96)(68,124,114,158)(69,139,115,86)(70,126,116,160)(71,141,117,88)(72,128,118,162), (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), (1,185,7,191)(2,184,8,190)(3,183,9,189)(4,182,10,188)(5,181,11,187)(6,192,12,186)(13,86,19,92)(14,85,20,91)(15,96,21,90)(16,95,22,89)(17,94,23,88)(18,93,24,87)(25,39,31,45)(26,38,32,44)(27,37,33,43)(28,48,34,42)(29,47,35,41)(30,46,36,40)(49,126,55,132)(50,125,56,131)(51,124,57,130)(52,123,58,129)(53,122,59,128)(54,121,60,127)(61,103,67,97)(62,102,68,108)(63,101,69,107)(64,100,70,106)(65,99,71,105)(66,98,72,104)(73,139,79,133)(74,138,80,144)(75,137,81,143)(76,136,82,142)(77,135,83,141)(78,134,84,140)(109,169,115,175)(110,180,116,174)(111,179,117,173)(112,178,118,172)(113,177,119,171)(114,176,120,170)(145,158,151,164)(146,157,152,163)(147,168,153,162)(148,167,154,161)(149,166,155,160)(150,165,156,159)>;
G:=Group( (1,71,44,117)(2,118,45,72)(3,61,46,119)(4,120,47,62)(5,63,48,109)(6,110,37,64)(7,65,38,111)(8,112,39,66)(9,67,40,113)(10,114,41,68)(11,69,42,115)(12,116,43,70)(13,50,73,156)(14,145,74,51)(15,52,75,146)(16,147,76,53)(17,54,77,148)(18,149,78,55)(19,56,79,150)(20,151,80,57)(21,58,81,152)(22,153,82,59)(23,60,83,154)(24,155,84,49)(25,104,184,172)(26,173,185,105)(27,106,186,174)(28,175,187,107)(29,108,188,176)(30,177,189,97)(31,98,190,178)(32,179,191,99)(33,100,192,180)(34,169,181,101)(35,102,182,170)(36,171,183,103)(85,158,138,124)(86,125,139,159)(87,160,140,126)(88,127,141,161)(89,162,142,128)(90,129,143,163)(91,164,144,130)(92,131,133,165)(93,166,134,132)(94,121,135,167)(95,168,136,122)(96,123,137,157), (1,161,44,127)(2,142,45,89)(3,163,46,129)(4,144,47,91)(5,165,48,131)(6,134,37,93)(7,167,38,121)(8,136,39,95)(9,157,40,123)(10,138,41,85)(11,159,42,125)(12,140,43,87)(13,169,73,101)(14,35,74,182)(15,171,75,103)(16,25,76,184)(17,173,77,105)(18,27,78,186)(19,175,79,107)(20,29,80,188)(21,177,81,97)(22,31,82,190)(23,179,83,99)(24,33,84,192)(26,148,185,54)(28,150,187,56)(30,152,189,58)(32,154,191,60)(34,156,181,50)(36,146,183,52)(49,100,155,180)(51,102,145,170)(53,104,147,172)(55,106,149,174)(57,108,151,176)(59,98,153,178)(61,143,119,90)(62,130,120,164)(63,133,109,92)(64,132,110,166)(65,135,111,94)(66,122,112,168)(67,137,113,96)(68,124,114,158)(69,139,115,86)(70,126,116,160)(71,141,117,88)(72,128,118,162), (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), (1,185,7,191)(2,184,8,190)(3,183,9,189)(4,182,10,188)(5,181,11,187)(6,192,12,186)(13,86,19,92)(14,85,20,91)(15,96,21,90)(16,95,22,89)(17,94,23,88)(18,93,24,87)(25,39,31,45)(26,38,32,44)(27,37,33,43)(28,48,34,42)(29,47,35,41)(30,46,36,40)(49,126,55,132)(50,125,56,131)(51,124,57,130)(52,123,58,129)(53,122,59,128)(54,121,60,127)(61,103,67,97)(62,102,68,108)(63,101,69,107)(64,100,70,106)(65,99,71,105)(66,98,72,104)(73,139,79,133)(74,138,80,144)(75,137,81,143)(76,136,82,142)(77,135,83,141)(78,134,84,140)(109,169,115,175)(110,180,116,174)(111,179,117,173)(112,178,118,172)(113,177,119,171)(114,176,120,170)(145,158,151,164)(146,157,152,163)(147,168,153,162)(148,167,154,161)(149,166,155,160)(150,165,156,159) );
G=PermutationGroup([[(1,71,44,117),(2,118,45,72),(3,61,46,119),(4,120,47,62),(5,63,48,109),(6,110,37,64),(7,65,38,111),(8,112,39,66),(9,67,40,113),(10,114,41,68),(11,69,42,115),(12,116,43,70),(13,50,73,156),(14,145,74,51),(15,52,75,146),(16,147,76,53),(17,54,77,148),(18,149,78,55),(19,56,79,150),(20,151,80,57),(21,58,81,152),(22,153,82,59),(23,60,83,154),(24,155,84,49),(25,104,184,172),(26,173,185,105),(27,106,186,174),(28,175,187,107),(29,108,188,176),(30,177,189,97),(31,98,190,178),(32,179,191,99),(33,100,192,180),(34,169,181,101),(35,102,182,170),(36,171,183,103),(85,158,138,124),(86,125,139,159),(87,160,140,126),(88,127,141,161),(89,162,142,128),(90,129,143,163),(91,164,144,130),(92,131,133,165),(93,166,134,132),(94,121,135,167),(95,168,136,122),(96,123,137,157)], [(1,161,44,127),(2,142,45,89),(3,163,46,129),(4,144,47,91),(5,165,48,131),(6,134,37,93),(7,167,38,121),(8,136,39,95),(9,157,40,123),(10,138,41,85),(11,159,42,125),(12,140,43,87),(13,169,73,101),(14,35,74,182),(15,171,75,103),(16,25,76,184),(17,173,77,105),(18,27,78,186),(19,175,79,107),(20,29,80,188),(21,177,81,97),(22,31,82,190),(23,179,83,99),(24,33,84,192),(26,148,185,54),(28,150,187,56),(30,152,189,58),(32,154,191,60),(34,156,181,50),(36,146,183,52),(49,100,155,180),(51,102,145,170),(53,104,147,172),(55,106,149,174),(57,108,151,176),(59,98,153,178),(61,143,119,90),(62,130,120,164),(63,133,109,92),(64,132,110,166),(65,135,111,94),(66,122,112,168),(67,137,113,96),(68,124,114,158),(69,139,115,86),(70,126,116,160),(71,141,117,88),(72,128,118,162)], [(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)], [(1,185,7,191),(2,184,8,190),(3,183,9,189),(4,182,10,188),(5,181,11,187),(6,192,12,186),(13,86,19,92),(14,85,20,91),(15,96,21,90),(16,95,22,89),(17,94,23,88),(18,93,24,87),(25,39,31,45),(26,38,32,44),(27,37,33,43),(28,48,34,42),(29,47,35,41),(30,46,36,40),(49,126,55,132),(50,125,56,131),(51,124,57,130),(52,123,58,129),(53,122,59,128),(54,121,60,127),(61,103,67,97),(62,102,68,108),(63,101,69,107),(64,100,70,106),(65,99,71,105),(66,98,72,104),(73,139,79,133),(74,138,80,144),(75,137,81,143),(76,136,82,142),(77,135,83,141),(78,134,84,140),(109,169,115,175),(110,180,116,174),(111,179,117,173),(112,178,118,172),(113,177,119,171),(114,176,120,170),(145,158,151,164),(146,157,152,163),(147,168,153,162),(148,167,154,161),(149,166,155,160),(150,165,156,159)]])
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 | Q16 | C4○D4 | Dic6 | C8⋊C22 | D4⋊2S3 | S3×D4 | Q8⋊3D6 | S3×Q16 |
| kernel | Q8⋊3Dic6 | C6.Q16 | Dic3⋊C8 | C24⋊1C4 | 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⋊3Dic6 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 71 |
| 0 | 0 | 1 | 72 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 19 | 13 |
| 0 | 0 | 62 | 54 |
| 14 | 7 | 0 | 0 |
| 66 | 7 | 0 | 0 |
| 0 | 0 | 13 | 49 |
| 0 | 0 | 1 | 60 |
| 71 | 53 | 0 | 0 |
| 55 | 2 | 0 | 0 |
| 0 | 0 | 1 | 71 |
| 0 | 0 | 1 | 72 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,71,72],[72,0,0,0,0,72,0,0,0,0,19,62,0,0,13,54],[14,66,0,0,7,7,0,0,0,0,13,1,0,0,49,60],[71,55,0,0,53,2,0,0,0,0,1,1,0,0,71,72] >;
Q8⋊3Dic6 in GAP, Magma, Sage, TeX
Q_8\rtimes_3{\rm Dic}_6 % in TeX
G:=Group("Q8:3Dic6"); // GroupNames label
G:=SmallGroup(192,352);
// by ID
G=gap.SmallGroup(192,352);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,232,254,219,226,851,438,102,6278]);
// 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,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations