metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8.4Dic6, C4⋊C4.21D6, C3⋊2(Q8.Q8), (C3×Q8).2Q8, C12.8(C2×Q8), (C2×C8).121D6, C8⋊Dic3.7C2, C4.8(C2×Dic6), C6.46(C4○D8), Dic3⋊C8.6C2, (C2×Q8).126D6, Q8⋊C4.6S3, (Q8×Dic3).6C2, C6.Q16.3C2, Q8⋊2Dic3.6C2, C22.193(S3×D4), C6.14(C22⋊Q8), C4.Dic6.3C2, (C6×Q8).22C22, C12.160(C4○D4), C4.85(D4⋊2S3), (C2×C24).132C22, (C2×C12).239C23, (C2×Dic3).151D4, C2.12(Q16⋊S3), C6.57(C8.C22), C4⋊Dic3.87C22, C2.15(Q8.7D6), (C4×Dic3).23C22, C2.19(Dic3.D4), (C2×C6).252(C2×D4), (C2×C3⋊C8).34C22, (C3×C4⋊C4).40C22, (C3×Q8⋊C4).6C2, (C2×C4).346(C22×S3), SmallGroup(192,358)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8.4Dic6
G = < a,b,c,d | a4=c12=1, b2=a2, d2=a2c6, bab-1=cac-1=a-1, ad=da, cbc-1=ab, bd=db, dcd-1=c-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, C6.Q16, Dic3⋊C8, C8⋊Dic3, Q8⋊2Dic3, C3×Q8⋊C4, C4.Dic6, Q8×Dic3, Q8.4Dic6
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, Q8.7D6, Q16⋊S3, Q8.4Dic6
►Regular action on 192 points
Generators in S192
(1 128 166 110)(2 111 167 129)(3 130 168 112)(4 113 157 131)(5 132 158 114)(6 115 159 121)(7 122 160 116)(8 117 161 123)(9 124 162 118)(10 119 163 125)(11 126 164 120)(12 109 165 127)(13 66 86 174)(14 175 87 67)(15 68 88 176)(16 177 89 69)(17 70 90 178)(18 179 91 71)(19 72 92 180)(20 169 93 61)(21 62 94 170)(22 171 95 63)(23 64 96 172)(24 173 85 65)(25 134 103 57)(26 58 104 135)(27 136 105 59)(28 60 106 137)(29 138 107 49)(30 50 108 139)(31 140 97 51)(32 52 98 141)(33 142 99 53)(34 54 100 143)(35 144 101 55)(36 56 102 133)(37 152 83 184)(38 185 84 153)(39 154 73 186)(40 187 74 155)(41 156 75 188)(42 189 76 145)(43 146 77 190)(44 191 78 147)(45 148 79 192)(46 181 80 149)(47 150 81 182)(48 183 82 151)
(1 29 166 107)(2 50 167 139)(3 31 168 97)(4 52 157 141)(5 33 158 99)(6 54 159 143)(7 35 160 101)(8 56 161 133)(9 25 162 103)(10 58 163 135)(11 27 164 105)(12 60 165 137)(13 186 86 154)(14 74 87 40)(15 188 88 156)(16 76 89 42)(17 190 90 146)(18 78 91 44)(19 192 92 148)(20 80 93 46)(21 182 94 150)(22 82 95 48)(23 184 96 152)(24 84 85 38)(26 125 104 119)(28 127 106 109)(30 129 108 111)(32 131 98 113)(34 121 100 115)(36 123 102 117)(37 64 83 172)(39 66 73 174)(41 68 75 176)(43 70 77 178)(45 72 79 180)(47 62 81 170)(49 110 138 128)(51 112 140 130)(53 114 142 132)(55 116 144 122)(57 118 134 124)(59 120 136 126)(61 149 169 181)(63 151 171 183)(65 153 173 185)(67 155 175 187)(69 145 177 189)(71 147 179 191)
(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)
(1 181 160 155)(2 192 161 154)(3 191 162 153)(4 190 163 152)(5 189 164 151)(6 188 165 150)(7 187 166 149)(8 186 167 148)(9 185 168 147)(10 184 157 146)(11 183 158 145)(12 182 159 156)(13 50 92 133)(14 49 93 144)(15 60 94 143)(16 59 95 142)(17 58 96 141)(18 57 85 140)(19 56 86 139)(20 55 87 138)(21 54 88 137)(22 53 89 136)(23 52 90 135)(24 51 91 134)(25 65 97 179)(26 64 98 178)(27 63 99 177)(28 62 100 176)(29 61 101 175)(30 72 102 174)(31 71 103 173)(32 70 104 172)(33 69 105 171)(34 68 106 170)(35 67 107 169)(36 66 108 180)(37 131 77 119)(38 130 78 118)(39 129 79 117)(40 128 80 116)(41 127 81 115)(42 126 82 114)(43 125 83 113)(44 124 84 112)(45 123 73 111)(46 122 74 110)(47 121 75 109)(48 132 76 120)
G:=sub<Sym(192)| (1,128,166,110)(2,111,167,129)(3,130,168,112)(4,113,157,131)(5,132,158,114)(6,115,159,121)(7,122,160,116)(8,117,161,123)(9,124,162,118)(10,119,163,125)(11,126,164,120)(12,109,165,127)(13,66,86,174)(14,175,87,67)(15,68,88,176)(16,177,89,69)(17,70,90,178)(18,179,91,71)(19,72,92,180)(20,169,93,61)(21,62,94,170)(22,171,95,63)(23,64,96,172)(24,173,85,65)(25,134,103,57)(26,58,104,135)(27,136,105,59)(28,60,106,137)(29,138,107,49)(30,50,108,139)(31,140,97,51)(32,52,98,141)(33,142,99,53)(34,54,100,143)(35,144,101,55)(36,56,102,133)(37,152,83,184)(38,185,84,153)(39,154,73,186)(40,187,74,155)(41,156,75,188)(42,189,76,145)(43,146,77,190)(44,191,78,147)(45,148,79,192)(46,181,80,149)(47,150,81,182)(48,183,82,151), (1,29,166,107)(2,50,167,139)(3,31,168,97)(4,52,157,141)(5,33,158,99)(6,54,159,143)(7,35,160,101)(8,56,161,133)(9,25,162,103)(10,58,163,135)(11,27,164,105)(12,60,165,137)(13,186,86,154)(14,74,87,40)(15,188,88,156)(16,76,89,42)(17,190,90,146)(18,78,91,44)(19,192,92,148)(20,80,93,46)(21,182,94,150)(22,82,95,48)(23,184,96,152)(24,84,85,38)(26,125,104,119)(28,127,106,109)(30,129,108,111)(32,131,98,113)(34,121,100,115)(36,123,102,117)(37,64,83,172)(39,66,73,174)(41,68,75,176)(43,70,77,178)(45,72,79,180)(47,62,81,170)(49,110,138,128)(51,112,140,130)(53,114,142,132)(55,116,144,122)(57,118,134,124)(59,120,136,126)(61,149,169,181)(63,151,171,183)(65,153,173,185)(67,155,175,187)(69,145,177,189)(71,147,179,191), (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), (1,181,160,155)(2,192,161,154)(3,191,162,153)(4,190,163,152)(5,189,164,151)(6,188,165,150)(7,187,166,149)(8,186,167,148)(9,185,168,147)(10,184,157,146)(11,183,158,145)(12,182,159,156)(13,50,92,133)(14,49,93,144)(15,60,94,143)(16,59,95,142)(17,58,96,141)(18,57,85,140)(19,56,86,139)(20,55,87,138)(21,54,88,137)(22,53,89,136)(23,52,90,135)(24,51,91,134)(25,65,97,179)(26,64,98,178)(27,63,99,177)(28,62,100,176)(29,61,101,175)(30,72,102,174)(31,71,103,173)(32,70,104,172)(33,69,105,171)(34,68,106,170)(35,67,107,169)(36,66,108,180)(37,131,77,119)(38,130,78,118)(39,129,79,117)(40,128,80,116)(41,127,81,115)(42,126,82,114)(43,125,83,113)(44,124,84,112)(45,123,73,111)(46,122,74,110)(47,121,75,109)(48,132,76,120)>;
G:=Group( (1,128,166,110)(2,111,167,129)(3,130,168,112)(4,113,157,131)(5,132,158,114)(6,115,159,121)(7,122,160,116)(8,117,161,123)(9,124,162,118)(10,119,163,125)(11,126,164,120)(12,109,165,127)(13,66,86,174)(14,175,87,67)(15,68,88,176)(16,177,89,69)(17,70,90,178)(18,179,91,71)(19,72,92,180)(20,169,93,61)(21,62,94,170)(22,171,95,63)(23,64,96,172)(24,173,85,65)(25,134,103,57)(26,58,104,135)(27,136,105,59)(28,60,106,137)(29,138,107,49)(30,50,108,139)(31,140,97,51)(32,52,98,141)(33,142,99,53)(34,54,100,143)(35,144,101,55)(36,56,102,133)(37,152,83,184)(38,185,84,153)(39,154,73,186)(40,187,74,155)(41,156,75,188)(42,189,76,145)(43,146,77,190)(44,191,78,147)(45,148,79,192)(46,181,80,149)(47,150,81,182)(48,183,82,151), (1,29,166,107)(2,50,167,139)(3,31,168,97)(4,52,157,141)(5,33,158,99)(6,54,159,143)(7,35,160,101)(8,56,161,133)(9,25,162,103)(10,58,163,135)(11,27,164,105)(12,60,165,137)(13,186,86,154)(14,74,87,40)(15,188,88,156)(16,76,89,42)(17,190,90,146)(18,78,91,44)(19,192,92,148)(20,80,93,46)(21,182,94,150)(22,82,95,48)(23,184,96,152)(24,84,85,38)(26,125,104,119)(28,127,106,109)(30,129,108,111)(32,131,98,113)(34,121,100,115)(36,123,102,117)(37,64,83,172)(39,66,73,174)(41,68,75,176)(43,70,77,178)(45,72,79,180)(47,62,81,170)(49,110,138,128)(51,112,140,130)(53,114,142,132)(55,116,144,122)(57,118,134,124)(59,120,136,126)(61,149,169,181)(63,151,171,183)(65,153,173,185)(67,155,175,187)(69,145,177,189)(71,147,179,191), (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), (1,181,160,155)(2,192,161,154)(3,191,162,153)(4,190,163,152)(5,189,164,151)(6,188,165,150)(7,187,166,149)(8,186,167,148)(9,185,168,147)(10,184,157,146)(11,183,158,145)(12,182,159,156)(13,50,92,133)(14,49,93,144)(15,60,94,143)(16,59,95,142)(17,58,96,141)(18,57,85,140)(19,56,86,139)(20,55,87,138)(21,54,88,137)(22,53,89,136)(23,52,90,135)(24,51,91,134)(25,65,97,179)(26,64,98,178)(27,63,99,177)(28,62,100,176)(29,61,101,175)(30,72,102,174)(31,71,103,173)(32,70,104,172)(33,69,105,171)(34,68,106,170)(35,67,107,169)(36,66,108,180)(37,131,77,119)(38,130,78,118)(39,129,79,117)(40,128,80,116)(41,127,81,115)(42,126,82,114)(43,125,83,113)(44,124,84,112)(45,123,73,111)(46,122,74,110)(47,121,75,109)(48,132,76,120) );
G=PermutationGroup([[(1,128,166,110),(2,111,167,129),(3,130,168,112),(4,113,157,131),(5,132,158,114),(6,115,159,121),(7,122,160,116),(8,117,161,123),(9,124,162,118),(10,119,163,125),(11,126,164,120),(12,109,165,127),(13,66,86,174),(14,175,87,67),(15,68,88,176),(16,177,89,69),(17,70,90,178),(18,179,91,71),(19,72,92,180),(20,169,93,61),(21,62,94,170),(22,171,95,63),(23,64,96,172),(24,173,85,65),(25,134,103,57),(26,58,104,135),(27,136,105,59),(28,60,106,137),(29,138,107,49),(30,50,108,139),(31,140,97,51),(32,52,98,141),(33,142,99,53),(34,54,100,143),(35,144,101,55),(36,56,102,133),(37,152,83,184),(38,185,84,153),(39,154,73,186),(40,187,74,155),(41,156,75,188),(42,189,76,145),(43,146,77,190),(44,191,78,147),(45,148,79,192),(46,181,80,149),(47,150,81,182),(48,183,82,151)], [(1,29,166,107),(2,50,167,139),(3,31,168,97),(4,52,157,141),(5,33,158,99),(6,54,159,143),(7,35,160,101),(8,56,161,133),(9,25,162,103),(10,58,163,135),(11,27,164,105),(12,60,165,137),(13,186,86,154),(14,74,87,40),(15,188,88,156),(16,76,89,42),(17,190,90,146),(18,78,91,44),(19,192,92,148),(20,80,93,46),(21,182,94,150),(22,82,95,48),(23,184,96,152),(24,84,85,38),(26,125,104,119),(28,127,106,109),(30,129,108,111),(32,131,98,113),(34,121,100,115),(36,123,102,117),(37,64,83,172),(39,66,73,174),(41,68,75,176),(43,70,77,178),(45,72,79,180),(47,62,81,170),(49,110,138,128),(51,112,140,130),(53,114,142,132),(55,116,144,122),(57,118,134,124),(59,120,136,126),(61,149,169,181),(63,151,171,183),(65,153,173,185),(67,155,175,187),(69,145,177,189),(71,147,179,191)], [(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)], [(1,181,160,155),(2,192,161,154),(3,191,162,153),(4,190,163,152),(5,189,164,151),(6,188,165,150),(7,187,166,149),(8,186,167,148),(9,185,168,147),(10,184,157,146),(11,183,158,145),(12,182,159,156),(13,50,92,133),(14,49,93,144),(15,60,94,143),(16,59,95,142),(17,58,96,141),(18,57,85,140),(19,56,86,139),(20,55,87,138),(21,54,88,137),(22,53,89,136),(23,52,90,135),(24,51,91,134),(25,65,97,179),(26,64,98,178),(27,63,99,177),(28,62,100,176),(29,61,101,175),(30,72,102,174),(31,71,103,173),(32,70,104,172),(33,69,105,171),(34,68,106,170),(35,67,107,169),(36,66,108,180),(37,131,77,119),(38,130,78,118),(39,129,79,117),(40,128,80,116),(41,127,81,115),(42,126,82,114),(43,125,83,113),(44,124,84,112),(45,123,73,111),(46,122,74,110),(47,121,75,109),(48,132,76,120)]])
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 | Q8.7D6 | Q16⋊S3 |
| kernel | Q8.4Dic6 | C6.Q16 | Dic3⋊C8 | C8⋊Dic3 | 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 Q8.4Dic6 ►in GL4(𝔽73) generated by
| 0 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 8 | 9 | 0 | 0 |
| 9 | 65 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 67 | 29 | 0 | 0 |
| 29 | 6 | 0 | 0 |
| 0 | 0 | 7 | 7 |
| 0 | 0 | 66 | 14 |
| 27 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 34 | 65 |
| 0 | 0 | 26 | 39 |
G:=sub<GL(4,GF(73))| [0,72,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[8,9,0,0,9,65,0,0,0,0,72,0,0,0,0,72],[67,29,0,0,29,6,0,0,0,0,7,66,0,0,7,14],[27,0,0,0,0,27,0,0,0,0,34,26,0,0,65,39] >;
Q8.4Dic6 in GAP, Magma, Sage, TeX
Q_8._4{\rm Dic}_6 % in TeX
G:=Group("Q8.4Dic6"); // GroupNames label
G:=SmallGroup(192,358);
// by ID
G=gap.SmallGroup(192,358);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,232,926,219,226,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^12=1,b^2=a^2,d^2=a^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