metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8.3Dic6, C4⋊C4.19D6, (C2×C8).15D6, C3⋊1(Q8.Q8), (C3×Q8).1Q8, C12.7(C2×Q8), C24⋊1C4.7C2, C4.7(C2×Dic6), C6.67(C4○D8), Dic3⋊C8.4C2, (C2×Q8).125D6, Q8⋊C4.4S3, (Q8×Dic3).5C2, (C2×C24).15C22, Q8⋊2Dic3.4C2, C22.190(S3×D4), C6.13(C22⋊Q8), C4.Dic6.1C2, C12.Q8.3C2, C12.159(C4○D4), (C6×Q8).19C22, C2.6(D24⋊C2), C4.84(D4⋊2S3), (C2×C12).236C23, (C2×Dic3).150D4, C2.15(D4.D6), C6.33(C8.C22), C4⋊Dic3.85C22, (C4×Dic3).20C22, C2.18(Dic3.D4), (C2×C6).249(C2×D4), (C2×C3⋊C8).31C22, (C3×C4⋊C4).37C22, (C3×Q8⋊C4).4C2, (C2×C4).343(C22×S3), SmallGroup(192,355)
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, bd=db, dcd-1=a2c-1 >
Subgroups: 216 in 90 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Q8, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×Q8, C3⋊C8, C24, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Q8, Q8⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×Q8, C42.C2, C2×C3⋊C8, C4×Dic3, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C3×C4⋊C4, C2×C24, C6×Q8, Q8.Q8, C12.Q8, Dic3⋊C8, C24⋊1C4, Q8⋊2Dic3, C3×Q8⋊C4, C4.Dic6, Q8×Dic3, Q8.3Dic6
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, Dic6, C22×S3, C22⋊Q8, C4○D8, C8.C22, C2×Dic6, S3×D4, D4⋊2S3, Q8.Q8, Dic3.D4, D4.D6, D24⋊C2, Q8.3Dic6
►Regular action on 192 points
Generators in S192
(1 73 56 124)(2 125 57 74)(3 75 58 126)(4 127 59 76)(5 77 60 128)(6 129 49 78)(7 79 50 130)(8 131 51 80)(9 81 52 132)(10 121 53 82)(11 83 54 122)(12 123 55 84)(13 174 164 139)(14 140 165 175)(15 176 166 141)(16 142 167 177)(17 178 168 143)(18 144 157 179)(19 180 158 133)(20 134 159 169)(21 170 160 135)(22 136 161 171)(23 172 162 137)(24 138 163 173)(25 109 41 183)(26 184 42 110)(27 111 43 185)(28 186 44 112)(29 113 45 187)(30 188 46 114)(31 115 47 189)(32 190 48 116)(33 117 37 191)(34 192 38 118)(35 119 39 181)(36 182 40 120)(61 105 87 153)(62 154 88 106)(63 107 89 155)(64 156 90 108)(65 97 91 145)(66 146 92 98)(67 99 93 147)(68 148 94 100)(69 101 95 149)(70 150 96 102)(71 103 85 151)(72 152 86 104)
(1 166 56 15)(2 177 57 142)(3 168 58 17)(4 179 59 144)(5 158 60 19)(6 169 49 134)(7 160 50 21)(8 171 51 136)(9 162 52 23)(10 173 53 138)(11 164 54 13)(12 175 55 140)(14 123 165 84)(16 125 167 74)(18 127 157 76)(20 129 159 78)(22 131 161 80)(24 121 163 82)(25 95 41 69)(26 102 42 150)(27 85 43 71)(28 104 44 152)(29 87 45 61)(30 106 46 154)(31 89 47 63)(32 108 48 156)(33 91 37 65)(34 98 38 146)(35 93 39 67)(36 100 40 148)(62 188 88 114)(64 190 90 116)(66 192 92 118)(68 182 94 120)(70 184 96 110)(72 186 86 112)(73 176 124 141)(75 178 126 143)(77 180 128 133)(79 170 130 135)(81 172 132 137)(83 174 122 139)(97 191 145 117)(99 181 147 119)(101 183 149 109)(103 185 151 111)(105 187 153 113)(107 189 155 115)
(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)
(1 155 7 149)(2 106 8 100)(3 153 9 147)(4 104 10 98)(5 151 11 145)(6 102 12 108)(13 191 19 185)(14 116 20 110)(15 189 21 183)(16 114 22 120)(17 187 23 181)(18 112 24 118)(25 176 31 170)(26 140 32 134)(27 174 33 180)(28 138 34 144)(29 172 35 178)(30 136 36 142)(37 133 43 139)(38 179 44 173)(39 143 45 137)(40 177 46 171)(41 141 47 135)(42 175 48 169)(49 150 55 156)(50 101 56 107)(51 148 57 154)(52 99 58 105)(53 146 59 152)(54 97 60 103)(61 81 67 75)(62 131 68 125)(63 79 69 73)(64 129 70 123)(65 77 71 83)(66 127 72 121)(74 88 80 94)(76 86 82 92)(78 96 84 90)(85 122 91 128)(87 132 93 126)(89 130 95 124)(109 166 115 160)(111 164 117 158)(113 162 119 168)(157 186 163 192)(159 184 165 190)(161 182 167 188)
G:=sub<Sym(192)| (1,73,56,124)(2,125,57,74)(3,75,58,126)(4,127,59,76)(5,77,60,128)(6,129,49,78)(7,79,50,130)(8,131,51,80)(9,81,52,132)(10,121,53,82)(11,83,54,122)(12,123,55,84)(13,174,164,139)(14,140,165,175)(15,176,166,141)(16,142,167,177)(17,178,168,143)(18,144,157,179)(19,180,158,133)(20,134,159,169)(21,170,160,135)(22,136,161,171)(23,172,162,137)(24,138,163,173)(25,109,41,183)(26,184,42,110)(27,111,43,185)(28,186,44,112)(29,113,45,187)(30,188,46,114)(31,115,47,189)(32,190,48,116)(33,117,37,191)(34,192,38,118)(35,119,39,181)(36,182,40,120)(61,105,87,153)(62,154,88,106)(63,107,89,155)(64,156,90,108)(65,97,91,145)(66,146,92,98)(67,99,93,147)(68,148,94,100)(69,101,95,149)(70,150,96,102)(71,103,85,151)(72,152,86,104), (1,166,56,15)(2,177,57,142)(3,168,58,17)(4,179,59,144)(5,158,60,19)(6,169,49,134)(7,160,50,21)(8,171,51,136)(9,162,52,23)(10,173,53,138)(11,164,54,13)(12,175,55,140)(14,123,165,84)(16,125,167,74)(18,127,157,76)(20,129,159,78)(22,131,161,80)(24,121,163,82)(25,95,41,69)(26,102,42,150)(27,85,43,71)(28,104,44,152)(29,87,45,61)(30,106,46,154)(31,89,47,63)(32,108,48,156)(33,91,37,65)(34,98,38,146)(35,93,39,67)(36,100,40,148)(62,188,88,114)(64,190,90,116)(66,192,92,118)(68,182,94,120)(70,184,96,110)(72,186,86,112)(73,176,124,141)(75,178,126,143)(77,180,128,133)(79,170,130,135)(81,172,132,137)(83,174,122,139)(97,191,145,117)(99,181,147,119)(101,183,149,109)(103,185,151,111)(105,187,153,113)(107,189,155,115), (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), (1,155,7,149)(2,106,8,100)(3,153,9,147)(4,104,10,98)(5,151,11,145)(6,102,12,108)(13,191,19,185)(14,116,20,110)(15,189,21,183)(16,114,22,120)(17,187,23,181)(18,112,24,118)(25,176,31,170)(26,140,32,134)(27,174,33,180)(28,138,34,144)(29,172,35,178)(30,136,36,142)(37,133,43,139)(38,179,44,173)(39,143,45,137)(40,177,46,171)(41,141,47,135)(42,175,48,169)(49,150,55,156)(50,101,56,107)(51,148,57,154)(52,99,58,105)(53,146,59,152)(54,97,60,103)(61,81,67,75)(62,131,68,125)(63,79,69,73)(64,129,70,123)(65,77,71,83)(66,127,72,121)(74,88,80,94)(76,86,82,92)(78,96,84,90)(85,122,91,128)(87,132,93,126)(89,130,95,124)(109,166,115,160)(111,164,117,158)(113,162,119,168)(157,186,163,192)(159,184,165,190)(161,182,167,188)>;
G:=Group( (1,73,56,124)(2,125,57,74)(3,75,58,126)(4,127,59,76)(5,77,60,128)(6,129,49,78)(7,79,50,130)(8,131,51,80)(9,81,52,132)(10,121,53,82)(11,83,54,122)(12,123,55,84)(13,174,164,139)(14,140,165,175)(15,176,166,141)(16,142,167,177)(17,178,168,143)(18,144,157,179)(19,180,158,133)(20,134,159,169)(21,170,160,135)(22,136,161,171)(23,172,162,137)(24,138,163,173)(25,109,41,183)(26,184,42,110)(27,111,43,185)(28,186,44,112)(29,113,45,187)(30,188,46,114)(31,115,47,189)(32,190,48,116)(33,117,37,191)(34,192,38,118)(35,119,39,181)(36,182,40,120)(61,105,87,153)(62,154,88,106)(63,107,89,155)(64,156,90,108)(65,97,91,145)(66,146,92,98)(67,99,93,147)(68,148,94,100)(69,101,95,149)(70,150,96,102)(71,103,85,151)(72,152,86,104), (1,166,56,15)(2,177,57,142)(3,168,58,17)(4,179,59,144)(5,158,60,19)(6,169,49,134)(7,160,50,21)(8,171,51,136)(9,162,52,23)(10,173,53,138)(11,164,54,13)(12,175,55,140)(14,123,165,84)(16,125,167,74)(18,127,157,76)(20,129,159,78)(22,131,161,80)(24,121,163,82)(25,95,41,69)(26,102,42,150)(27,85,43,71)(28,104,44,152)(29,87,45,61)(30,106,46,154)(31,89,47,63)(32,108,48,156)(33,91,37,65)(34,98,38,146)(35,93,39,67)(36,100,40,148)(62,188,88,114)(64,190,90,116)(66,192,92,118)(68,182,94,120)(70,184,96,110)(72,186,86,112)(73,176,124,141)(75,178,126,143)(77,180,128,133)(79,170,130,135)(81,172,132,137)(83,174,122,139)(97,191,145,117)(99,181,147,119)(101,183,149,109)(103,185,151,111)(105,187,153,113)(107,189,155,115), (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), (1,155,7,149)(2,106,8,100)(3,153,9,147)(4,104,10,98)(5,151,11,145)(6,102,12,108)(13,191,19,185)(14,116,20,110)(15,189,21,183)(16,114,22,120)(17,187,23,181)(18,112,24,118)(25,176,31,170)(26,140,32,134)(27,174,33,180)(28,138,34,144)(29,172,35,178)(30,136,36,142)(37,133,43,139)(38,179,44,173)(39,143,45,137)(40,177,46,171)(41,141,47,135)(42,175,48,169)(49,150,55,156)(50,101,56,107)(51,148,57,154)(52,99,58,105)(53,146,59,152)(54,97,60,103)(61,81,67,75)(62,131,68,125)(63,79,69,73)(64,129,70,123)(65,77,71,83)(66,127,72,121)(74,88,80,94)(76,86,82,92)(78,96,84,90)(85,122,91,128)(87,132,93,126)(89,130,95,124)(109,166,115,160)(111,164,117,158)(113,162,119,168)(157,186,163,192)(159,184,165,190)(161,182,167,188) );
G=PermutationGroup([[(1,73,56,124),(2,125,57,74),(3,75,58,126),(4,127,59,76),(5,77,60,128),(6,129,49,78),(7,79,50,130),(8,131,51,80),(9,81,52,132),(10,121,53,82),(11,83,54,122),(12,123,55,84),(13,174,164,139),(14,140,165,175),(15,176,166,141),(16,142,167,177),(17,178,168,143),(18,144,157,179),(19,180,158,133),(20,134,159,169),(21,170,160,135),(22,136,161,171),(23,172,162,137),(24,138,163,173),(25,109,41,183),(26,184,42,110),(27,111,43,185),(28,186,44,112),(29,113,45,187),(30,188,46,114),(31,115,47,189),(32,190,48,116),(33,117,37,191),(34,192,38,118),(35,119,39,181),(36,182,40,120),(61,105,87,153),(62,154,88,106),(63,107,89,155),(64,156,90,108),(65,97,91,145),(66,146,92,98),(67,99,93,147),(68,148,94,100),(69,101,95,149),(70,150,96,102),(71,103,85,151),(72,152,86,104)], [(1,166,56,15),(2,177,57,142),(3,168,58,17),(4,179,59,144),(5,158,60,19),(6,169,49,134),(7,160,50,21),(8,171,51,136),(9,162,52,23),(10,173,53,138),(11,164,54,13),(12,175,55,140),(14,123,165,84),(16,125,167,74),(18,127,157,76),(20,129,159,78),(22,131,161,80),(24,121,163,82),(25,95,41,69),(26,102,42,150),(27,85,43,71),(28,104,44,152),(29,87,45,61),(30,106,46,154),(31,89,47,63),(32,108,48,156),(33,91,37,65),(34,98,38,146),(35,93,39,67),(36,100,40,148),(62,188,88,114),(64,190,90,116),(66,192,92,118),(68,182,94,120),(70,184,96,110),(72,186,86,112),(73,176,124,141),(75,178,126,143),(77,180,128,133),(79,170,130,135),(81,172,132,137),(83,174,122,139),(97,191,145,117),(99,181,147,119),(101,183,149,109),(103,185,151,111),(105,187,153,113),(107,189,155,115)], [(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)], [(1,155,7,149),(2,106,8,100),(3,153,9,147),(4,104,10,98),(5,151,11,145),(6,102,12,108),(13,191,19,185),(14,116,20,110),(15,189,21,183),(16,114,22,120),(17,187,23,181),(18,112,24,118),(25,176,31,170),(26,140,32,134),(27,174,33,180),(28,138,34,144),(29,172,35,178),(30,136,36,142),(37,133,43,139),(38,179,44,173),(39,143,45,137),(40,177,46,171),(41,141,47,135),(42,175,48,169),(49,150,55,156),(50,101,56,107),(51,148,57,154),(52,99,58,105),(53,146,59,152),(54,97,60,103),(61,81,67,75),(62,131,68,125),(63,79,69,73),(64,129,70,123),(65,77,71,83),(66,127,72,121),(74,88,80,94),(76,86,82,92),(78,96,84,90),(85,122,91,128),(87,132,93,126),(89,130,95,124),(109,166,115,160),(111,164,117,158),(113,162,119,168),(157,186,163,192),(159,184,165,190),(161,182,167,188)]])
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 | C4○D4 | Dic6 | C4○D8 | C8.C22 | D4⋊2S3 | S3×D4 | D4.D6 | D24⋊C2 |
| kernel | Q8.3Dic6 | C12.Q8 | Dic3⋊C8 | C24⋊1C4 | Q8⋊2Dic3 | C3×Q8⋊C4 | C4.Dic6 | Q8×Dic3 | Q8⋊C4 | C2×Dic3 | C3×Q8 | C4⋊C4 | C2×C8 | C2×Q8 | C12 | Q8 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 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 | 3 |
| 0 | 0 | 48 | 72 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 62 | 37 |
| 0 | 0 | 48 | 11 |
| 66 | 66 | 0 | 0 |
| 7 | 59 | 0 | 0 |
| 0 | 0 | 41 | 17 |
| 0 | 0 | 17 | 32 |
| 36 | 48 | 0 | 0 |
| 11 | 37 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 46 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,1,48,0,0,3,72],[72,0,0,0,0,72,0,0,0,0,62,48,0,0,37,11],[66,7,0,0,66,59,0,0,0,0,41,17,0,0,17,32],[36,11,0,0,48,37,0,0,0,0,46,0,0,0,0,46] >;
Q8.3Dic6 in GAP, Magma, Sage, TeX
Q_8._3{\rm Dic}_6 % in TeX
G:=Group("Q8.3Dic6"); // GroupNames label
G:=SmallGroup(192,355);
// by ID
G=gap.SmallGroup(192,355);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,232,926,219,58,851,438,102,6278]);
// 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=a^2*c^-1>;
// generators/relations