metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6.2Q8, C4.6(S3×Q8), C4⋊C4.45D6, (C2×C8).26D6, C3⋊5(Q8.Q8), C2.D8.6S3, C12.19(C2×Q8), C6.27(C4○D8), Dic3⋊C8.8C2, C4.79(C4○D12), (C2×Dic3).47D4, C6.SD16.9C2, C22.226(S3×D4), C6.40(C22⋊Q8), C12.Q8.8C2, C2.Dic12.7C2, C4.Dic6.8C2, C12.171(C4○D4), C2.12(D8⋊3S3), (C2×C24).168C22, (C2×C12).293C23, C2.17(D6⋊Q8), Dic6⋊C4.9C2, C2.21(Q16⋊S3), C6.68(C8.C22), C4⋊Dic3.119C22, (C2×Dic6).88C22, (C4×Dic3).36C22, (C2×C6).298(C2×D4), (C2×C3⋊C8).67C22, (C3×C2.D8).12C2, (C3×C4⋊C4).86C22, (C2×C4).396(C22×S3), SmallGroup(192,436)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic6.2Q8
G = < a,b,c,d | a12=c4=1, b2=a6, d2=a6c2, bab-1=a-1, cac-1=a7, dad-1=a5, cbc-1=a3b, bd=db, dcd-1=c-1 >
Subgroups: 224 in 90 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×Q8, C3⋊C8, C24, Dic6, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, 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, C2×Dic6, Q8.Q8, C12.Q8, C6.SD16, Dic3⋊C8, C2.Dic12, C3×C2.D8, Dic6⋊C4, C4.Dic6, Dic6.2Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, C22×S3, C22⋊Q8, C4○D8, C8.C22, C4○D12, S3×D4, S3×Q8, Q8.Q8, D6⋊Q8, D8⋊3S3, Q16⋊S3, Dic6.2Q8
►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)
(1 163 7 157)(2 162 8 168)(3 161 9 167)(4 160 10 166)(5 159 11 165)(6 158 12 164)(13 148 19 154)(14 147 20 153)(15 146 21 152)(16 145 22 151)(17 156 23 150)(18 155 24 149)(25 130 31 124)(26 129 32 123)(27 128 33 122)(28 127 34 121)(29 126 35 132)(30 125 36 131)(37 191 43 185)(38 190 44 184)(39 189 45 183)(40 188 46 182)(41 187 47 181)(42 186 48 192)(49 135 55 141)(50 134 56 140)(51 133 57 139)(52 144 58 138)(53 143 59 137)(54 142 60 136)(61 170 67 176)(62 169 68 175)(63 180 69 174)(64 179 70 173)(65 178 71 172)(66 177 72 171)(73 109 79 115)(74 120 80 114)(75 119 81 113)(76 118 82 112)(77 117 83 111)(78 116 84 110)(85 108 91 102)(86 107 92 101)(87 106 93 100)(88 105 94 99)(89 104 95 98)(90 103 96 97)
(1 22 129 58)(2 17 130 53)(3 24 131 60)(4 19 132 55)(5 14 121 50)(6 21 122 57)(7 16 123 52)(8 23 124 59)(9 18 125 54)(10 13 126 49)(11 20 127 56)(12 15 128 51)(25 134 168 147)(26 141 157 154)(27 136 158 149)(28 143 159 156)(29 138 160 151)(30 133 161 146)(31 140 162 153)(32 135 163 148)(33 142 164 155)(34 137 165 150)(35 144 166 145)(36 139 167 152)(37 90 110 170)(38 85 111 177)(39 92 112 172)(40 87 113 179)(41 94 114 174)(42 89 115 169)(43 96 116 176)(44 91 117 171)(45 86 118 178)(46 93 119 173)(47 88 120 180)(48 95 109 175)(61 182 97 81)(62 189 98 76)(63 184 99 83)(64 191 100 78)(65 186 101 73)(66 181 102 80)(67 188 103 75)(68 183 104 82)(69 190 105 77)(70 185 106 84)(71 192 107 79)(72 187 108 74)
(1 87 123 173)(2 92 124 178)(3 85 125 171)(4 90 126 176)(5 95 127 169)(6 88 128 174)(7 93 129 179)(8 86 130 172)(9 91 131 177)(10 96 132 170)(11 89 121 175)(12 94 122 180)(13 43 55 110)(14 48 56 115)(15 41 57 120)(16 46 58 113)(17 39 59 118)(18 44 60 111)(19 37 49 116)(20 42 50 109)(21 47 51 114)(22 40 52 119)(23 45 53 112)(24 38 54 117)(25 71 162 101)(26 64 163 106)(27 69 164 99)(28 62 165 104)(29 67 166 97)(30 72 167 102)(31 65 168 107)(32 70 157 100)(33 63 158 105)(34 68 159 98)(35 61 160 103)(36 66 161 108)(73 147 192 140)(74 152 181 133)(75 145 182 138)(76 150 183 143)(77 155 184 136)(78 148 185 141)(79 153 186 134)(80 146 187 139)(81 151 188 144)(82 156 189 137)(83 149 190 142)(84 154 191 135)
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), (1,163,7,157)(2,162,8,168)(3,161,9,167)(4,160,10,166)(5,159,11,165)(6,158,12,164)(13,148,19,154)(14,147,20,153)(15,146,21,152)(16,145,22,151)(17,156,23,150)(18,155,24,149)(25,130,31,124)(26,129,32,123)(27,128,33,122)(28,127,34,121)(29,126,35,132)(30,125,36,131)(37,191,43,185)(38,190,44,184)(39,189,45,183)(40,188,46,182)(41,187,47,181)(42,186,48,192)(49,135,55,141)(50,134,56,140)(51,133,57,139)(52,144,58,138)(53,143,59,137)(54,142,60,136)(61,170,67,176)(62,169,68,175)(63,180,69,174)(64,179,70,173)(65,178,71,172)(66,177,72,171)(73,109,79,115)(74,120,80,114)(75,119,81,113)(76,118,82,112)(77,117,83,111)(78,116,84,110)(85,108,91,102)(86,107,92,101)(87,106,93,100)(88,105,94,99)(89,104,95,98)(90,103,96,97), (1,22,129,58)(2,17,130,53)(3,24,131,60)(4,19,132,55)(5,14,121,50)(6,21,122,57)(7,16,123,52)(8,23,124,59)(9,18,125,54)(10,13,126,49)(11,20,127,56)(12,15,128,51)(25,134,168,147)(26,141,157,154)(27,136,158,149)(28,143,159,156)(29,138,160,151)(30,133,161,146)(31,140,162,153)(32,135,163,148)(33,142,164,155)(34,137,165,150)(35,144,166,145)(36,139,167,152)(37,90,110,170)(38,85,111,177)(39,92,112,172)(40,87,113,179)(41,94,114,174)(42,89,115,169)(43,96,116,176)(44,91,117,171)(45,86,118,178)(46,93,119,173)(47,88,120,180)(48,95,109,175)(61,182,97,81)(62,189,98,76)(63,184,99,83)(64,191,100,78)(65,186,101,73)(66,181,102,80)(67,188,103,75)(68,183,104,82)(69,190,105,77)(70,185,106,84)(71,192,107,79)(72,187,108,74), (1,87,123,173)(2,92,124,178)(3,85,125,171)(4,90,126,176)(5,95,127,169)(6,88,128,174)(7,93,129,179)(8,86,130,172)(9,91,131,177)(10,96,132,170)(11,89,121,175)(12,94,122,180)(13,43,55,110)(14,48,56,115)(15,41,57,120)(16,46,58,113)(17,39,59,118)(18,44,60,111)(19,37,49,116)(20,42,50,109)(21,47,51,114)(22,40,52,119)(23,45,53,112)(24,38,54,117)(25,71,162,101)(26,64,163,106)(27,69,164,99)(28,62,165,104)(29,67,166,97)(30,72,167,102)(31,65,168,107)(32,70,157,100)(33,63,158,105)(34,68,159,98)(35,61,160,103)(36,66,161,108)(73,147,192,140)(74,152,181,133)(75,145,182,138)(76,150,183,143)(77,155,184,136)(78,148,185,141)(79,153,186,134)(80,146,187,139)(81,151,188,144)(82,156,189,137)(83,149,190,142)(84,154,191,135)>;
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), (1,163,7,157)(2,162,8,168)(3,161,9,167)(4,160,10,166)(5,159,11,165)(6,158,12,164)(13,148,19,154)(14,147,20,153)(15,146,21,152)(16,145,22,151)(17,156,23,150)(18,155,24,149)(25,130,31,124)(26,129,32,123)(27,128,33,122)(28,127,34,121)(29,126,35,132)(30,125,36,131)(37,191,43,185)(38,190,44,184)(39,189,45,183)(40,188,46,182)(41,187,47,181)(42,186,48,192)(49,135,55,141)(50,134,56,140)(51,133,57,139)(52,144,58,138)(53,143,59,137)(54,142,60,136)(61,170,67,176)(62,169,68,175)(63,180,69,174)(64,179,70,173)(65,178,71,172)(66,177,72,171)(73,109,79,115)(74,120,80,114)(75,119,81,113)(76,118,82,112)(77,117,83,111)(78,116,84,110)(85,108,91,102)(86,107,92,101)(87,106,93,100)(88,105,94,99)(89,104,95,98)(90,103,96,97), (1,22,129,58)(2,17,130,53)(3,24,131,60)(4,19,132,55)(5,14,121,50)(6,21,122,57)(7,16,123,52)(8,23,124,59)(9,18,125,54)(10,13,126,49)(11,20,127,56)(12,15,128,51)(25,134,168,147)(26,141,157,154)(27,136,158,149)(28,143,159,156)(29,138,160,151)(30,133,161,146)(31,140,162,153)(32,135,163,148)(33,142,164,155)(34,137,165,150)(35,144,166,145)(36,139,167,152)(37,90,110,170)(38,85,111,177)(39,92,112,172)(40,87,113,179)(41,94,114,174)(42,89,115,169)(43,96,116,176)(44,91,117,171)(45,86,118,178)(46,93,119,173)(47,88,120,180)(48,95,109,175)(61,182,97,81)(62,189,98,76)(63,184,99,83)(64,191,100,78)(65,186,101,73)(66,181,102,80)(67,188,103,75)(68,183,104,82)(69,190,105,77)(70,185,106,84)(71,192,107,79)(72,187,108,74), (1,87,123,173)(2,92,124,178)(3,85,125,171)(4,90,126,176)(5,95,127,169)(6,88,128,174)(7,93,129,179)(8,86,130,172)(9,91,131,177)(10,96,132,170)(11,89,121,175)(12,94,122,180)(13,43,55,110)(14,48,56,115)(15,41,57,120)(16,46,58,113)(17,39,59,118)(18,44,60,111)(19,37,49,116)(20,42,50,109)(21,47,51,114)(22,40,52,119)(23,45,53,112)(24,38,54,117)(25,71,162,101)(26,64,163,106)(27,69,164,99)(28,62,165,104)(29,67,166,97)(30,72,167,102)(31,65,168,107)(32,70,157,100)(33,63,158,105)(34,68,159,98)(35,61,160,103)(36,66,161,108)(73,147,192,140)(74,152,181,133)(75,145,182,138)(76,150,183,143)(77,155,184,136)(78,148,185,141)(79,153,186,134)(80,146,187,139)(81,151,188,144)(82,156,189,137)(83,149,190,142)(84,154,191,135) );
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)], [(1,163,7,157),(2,162,8,168),(3,161,9,167),(4,160,10,166),(5,159,11,165),(6,158,12,164),(13,148,19,154),(14,147,20,153),(15,146,21,152),(16,145,22,151),(17,156,23,150),(18,155,24,149),(25,130,31,124),(26,129,32,123),(27,128,33,122),(28,127,34,121),(29,126,35,132),(30,125,36,131),(37,191,43,185),(38,190,44,184),(39,189,45,183),(40,188,46,182),(41,187,47,181),(42,186,48,192),(49,135,55,141),(50,134,56,140),(51,133,57,139),(52,144,58,138),(53,143,59,137),(54,142,60,136),(61,170,67,176),(62,169,68,175),(63,180,69,174),(64,179,70,173),(65,178,71,172),(66,177,72,171),(73,109,79,115),(74,120,80,114),(75,119,81,113),(76,118,82,112),(77,117,83,111),(78,116,84,110),(85,108,91,102),(86,107,92,101),(87,106,93,100),(88,105,94,99),(89,104,95,98),(90,103,96,97)], [(1,22,129,58),(2,17,130,53),(3,24,131,60),(4,19,132,55),(5,14,121,50),(6,21,122,57),(7,16,123,52),(8,23,124,59),(9,18,125,54),(10,13,126,49),(11,20,127,56),(12,15,128,51),(25,134,168,147),(26,141,157,154),(27,136,158,149),(28,143,159,156),(29,138,160,151),(30,133,161,146),(31,140,162,153),(32,135,163,148),(33,142,164,155),(34,137,165,150),(35,144,166,145),(36,139,167,152),(37,90,110,170),(38,85,111,177),(39,92,112,172),(40,87,113,179),(41,94,114,174),(42,89,115,169),(43,96,116,176),(44,91,117,171),(45,86,118,178),(46,93,119,173),(47,88,120,180),(48,95,109,175),(61,182,97,81),(62,189,98,76),(63,184,99,83),(64,191,100,78),(65,186,101,73),(66,181,102,80),(67,188,103,75),(68,183,104,82),(69,190,105,77),(70,185,106,84),(71,192,107,79),(72,187,108,74)], [(1,87,123,173),(2,92,124,178),(3,85,125,171),(4,90,126,176),(5,95,127,169),(6,88,128,174),(7,93,129,179),(8,86,130,172),(9,91,131,177),(10,96,132,170),(11,89,121,175),(12,94,122,180),(13,43,55,110),(14,48,56,115),(15,41,57,120),(16,46,58,113),(17,39,59,118),(18,44,60,111),(19,37,49,116),(20,42,50,109),(21,47,51,114),(22,40,52,119),(23,45,53,112),(24,38,54,117),(25,71,162,101),(26,64,163,106),(27,69,164,99),(28,62,165,104),(29,67,166,97),(30,72,167,102),(31,65,168,107),(32,70,157,100),(33,63,158,105),(34,68,159,98),(35,61,160,103),(36,66,161,108),(73,147,192,140),(74,152,181,133),(75,145,182,138),(76,150,183,143),(77,155,184,136),(78,148,185,141),(79,153,186,134),(80,146,187,139),(81,151,188,144),(82,156,189,137),(83,149,190,142),(84,154,191,135)]])
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 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | - | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D4 | D6 | D6 | C4○D4 | C4○D8 | C4○D12 | C8.C22 | S3×Q8 | S3×D4 | D8⋊3S3 | Q16⋊S3 |
| kernel | Dic6.2Q8 | C12.Q8 | C6.SD16 | Dic3⋊C8 | C2.Dic12 | C3×C2.D8 | Dic6⋊C4 | C4.Dic6 | C2.D8 | Dic6 | C2×Dic3 | C4⋊C4 | C2×C8 | C12 | C6 | C4 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 2 | 4 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of Dic6.2Q8 ►in GL4(𝔽73) generated by
| 9 | 0 | 0 | 0 |
| 22 | 65 | 0 | 0 |
| 0 | 0 | 72 | 63 |
| 0 | 0 | 44 | 1 |
| 16 | 54 | 0 | 0 |
| 48 | 57 | 0 | 0 |
| 0 | 0 | 11 | 63 |
| 0 | 0 | 56 | 62 |
| 27 | 0 | 0 | 0 |
| 57 | 46 | 0 | 0 |
| 0 | 0 | 71 | 3 |
| 0 | 0 | 72 | 2 |
| 67 | 71 | 0 | 0 |
| 55 | 6 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 46 |
G:=sub<GL(4,GF(73))| [9,22,0,0,0,65,0,0,0,0,72,44,0,0,63,1],[16,48,0,0,54,57,0,0,0,0,11,56,0,0,63,62],[27,57,0,0,0,46,0,0,0,0,71,72,0,0,3,2],[67,55,0,0,71,6,0,0,0,0,46,0,0,0,0,46] >;
Dic6.2Q8 in GAP, Magma, Sage, TeX
{\rm Dic}_6._2Q_8 % in TeX
G:=Group("Dic6.2Q8"); // GroupNames label
G:=SmallGroup(192,436);
// by ID
G=gap.SmallGroup(192,436);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,344,1094,135,268,570,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=c^4=1,b^2=a^6,d^2=a^6*c^2,b*a*b^-1=a^-1,c*a*c^-1=a^7,d*a*d^-1=a^5,c*b*c^-1=a^3*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations