metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6⋊1Q8, Dic3.7SD16, C4.1(S3×Q8), C4⋊C4.31D6, C12⋊Q8.4C2, C12.9(C2×Q8), C4.Q8.4S3, C3⋊3(Q8⋊Q8), (C2×C8).135D6, C2.20(S3×SD16), C6.35(C2×SD16), C4.70(C4○D12), Dic3⋊C8.13C2, C6.SD16.5C2, C22.209(S3×D4), C6.35(C22⋊Q8), C12.Q8.4C2, C12.166(C4○D4), (C2×C12).270C23, (C2×C24).282C22, C2.12(D6⋊Q8), Dic6⋊C4.5C2, (C2×Dic3).161D4, C2.20(D4.D6), C6.38(C8.C22), C2.Dic12.13C2, C4⋊Dic3.102C22, (C4×Dic3).28C22, (C2×Dic6).80C22, (C3×C4.Q8).9C2, (C2×C6).275(C2×D4), (C2×C3⋊C8).52C22, (C3×C4⋊C4).63C22, (C2×C4).373(C22×S3), SmallGroup(192,413)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic6⋊Q8
G = < a,b,c,d | a12=c4=1, b2=a6, d2=c2, bab-1=a-1, cac-1=a5, dad-1=a7, bc=cb, dbd-1=a3b, dcd-1=c-1 >
Subgroups: 256 in 96 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Dic3, 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, C4.Q8, C4×Q8, C4⋊Q8, C2×C3⋊C8, C4×Dic3, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C2×C24, C2×Dic6, C2×Dic6, Q8⋊Q8, C12.Q8, C6.SD16, Dic3⋊C8, C2.Dic12, C3×C4.Q8, Dic6⋊C4, C12⋊Q8, Dic6⋊Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, SD16, C2×D4, C2×Q8, C4○D4, C22×S3, C22⋊Q8, C2×SD16, C8.C22, C4○D12, S3×D4, S3×Q8, Q8⋊Q8, D6⋊Q8, S3×SD16, D4.D6, Dic6⋊Q8
►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 25 7 31)(2 36 8 30)(3 35 9 29)(4 34 10 28)(5 33 11 27)(6 32 12 26)(13 192 19 186)(14 191 20 185)(15 190 21 184)(16 189 22 183)(17 188 23 182)(18 187 24 181)(37 53 43 59)(38 52 44 58)(39 51 45 57)(40 50 46 56)(41 49 47 55)(42 60 48 54)(61 174 67 180)(62 173 68 179)(63 172 69 178)(64 171 70 177)(65 170 71 176)(66 169 72 175)(73 138 79 144)(74 137 80 143)(75 136 81 142)(76 135 82 141)(77 134 83 140)(78 133 84 139)(85 101 91 107)(86 100 92 106)(87 99 93 105)(88 98 94 104)(89 97 95 103)(90 108 96 102)(109 167 115 161)(110 166 116 160)(111 165 117 159)(112 164 118 158)(113 163 119 157)(114 162 120 168)(121 154 127 148)(122 153 128 147)(123 152 129 146)(124 151 130 145)(125 150 131 156)(126 149 132 155)
(1 182 74 119)(2 187 75 112)(3 192 76 117)(4 185 77 110)(5 190 78 115)(6 183 79 120)(7 188 80 113)(8 181 81 118)(9 186 82 111)(10 191 83 116)(11 184 84 109)(12 189 73 114)(13 141 165 29)(14 134 166 34)(15 139 167 27)(16 144 168 32)(17 137 157 25)(18 142 158 30)(19 135 159 35)(20 140 160 28)(21 133 161 33)(22 138 162 26)(23 143 163 31)(24 136 164 36)(37 170 146 104)(38 175 147 97)(39 180 148 102)(40 173 149 107)(41 178 150 100)(42 171 151 105)(43 176 152 98)(44 169 153 103)(45 174 154 108)(46 179 155 101)(47 172 156 106)(48 177 145 99)(49 63 131 92)(50 68 132 85)(51 61 121 90)(52 66 122 95)(53 71 123 88)(54 64 124 93)(55 69 125 86)(56 62 126 91)(57 67 127 96)(58 72 128 89)(59 65 129 94)(60 70 130 87)
(1 108 74 174)(2 103 75 169)(3 98 76 176)(4 105 77 171)(5 100 78 178)(6 107 79 173)(7 102 80 180)(8 97 81 175)(9 104 82 170)(10 99 83 177)(11 106 84 172)(12 101 73 179)(13 132 165 50)(14 127 166 57)(15 122 167 52)(16 129 168 59)(17 124 157 54)(18 131 158 49)(19 126 159 56)(20 121 160 51)(21 128 161 58)(22 123 162 53)(23 130 163 60)(24 125 164 55)(25 93 137 64)(26 88 138 71)(27 95 139 66)(28 90 140 61)(29 85 141 68)(30 92 142 63)(31 87 143 70)(32 94 144 65)(33 89 133 72)(34 96 134 67)(35 91 135 62)(36 86 136 69)(37 186 146 111)(38 181 147 118)(39 188 148 113)(40 183 149 120)(41 190 150 115)(42 185 151 110)(43 192 152 117)(44 187 153 112)(45 182 154 119)(46 189 155 114)(47 184 156 109)(48 191 145 116)
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), (1,25,7,31)(2,36,8,30)(3,35,9,29)(4,34,10,28)(5,33,11,27)(6,32,12,26)(13,192,19,186)(14,191,20,185)(15,190,21,184)(16,189,22,183)(17,188,23,182)(18,187,24,181)(37,53,43,59)(38,52,44,58)(39,51,45,57)(40,50,46,56)(41,49,47,55)(42,60,48,54)(61,174,67,180)(62,173,68,179)(63,172,69,178)(64,171,70,177)(65,170,71,176)(66,169,72,175)(73,138,79,144)(74,137,80,143)(75,136,81,142)(76,135,82,141)(77,134,83,140)(78,133,84,139)(85,101,91,107)(86,100,92,106)(87,99,93,105)(88,98,94,104)(89,97,95,103)(90,108,96,102)(109,167,115,161)(110,166,116,160)(111,165,117,159)(112,164,118,158)(113,163,119,157)(114,162,120,168)(121,154,127,148)(122,153,128,147)(123,152,129,146)(124,151,130,145)(125,150,131,156)(126,149,132,155), (1,182,74,119)(2,187,75,112)(3,192,76,117)(4,185,77,110)(5,190,78,115)(6,183,79,120)(7,188,80,113)(8,181,81,118)(9,186,82,111)(10,191,83,116)(11,184,84,109)(12,189,73,114)(13,141,165,29)(14,134,166,34)(15,139,167,27)(16,144,168,32)(17,137,157,25)(18,142,158,30)(19,135,159,35)(20,140,160,28)(21,133,161,33)(22,138,162,26)(23,143,163,31)(24,136,164,36)(37,170,146,104)(38,175,147,97)(39,180,148,102)(40,173,149,107)(41,178,150,100)(42,171,151,105)(43,176,152,98)(44,169,153,103)(45,174,154,108)(46,179,155,101)(47,172,156,106)(48,177,145,99)(49,63,131,92)(50,68,132,85)(51,61,121,90)(52,66,122,95)(53,71,123,88)(54,64,124,93)(55,69,125,86)(56,62,126,91)(57,67,127,96)(58,72,128,89)(59,65,129,94)(60,70,130,87), (1,108,74,174)(2,103,75,169)(3,98,76,176)(4,105,77,171)(5,100,78,178)(6,107,79,173)(7,102,80,180)(8,97,81,175)(9,104,82,170)(10,99,83,177)(11,106,84,172)(12,101,73,179)(13,132,165,50)(14,127,166,57)(15,122,167,52)(16,129,168,59)(17,124,157,54)(18,131,158,49)(19,126,159,56)(20,121,160,51)(21,128,161,58)(22,123,162,53)(23,130,163,60)(24,125,164,55)(25,93,137,64)(26,88,138,71)(27,95,139,66)(28,90,140,61)(29,85,141,68)(30,92,142,63)(31,87,143,70)(32,94,144,65)(33,89,133,72)(34,96,134,67)(35,91,135,62)(36,86,136,69)(37,186,146,111)(38,181,147,118)(39,188,148,113)(40,183,149,120)(41,190,150,115)(42,185,151,110)(43,192,152,117)(44,187,153,112)(45,182,154,119)(46,189,155,114)(47,184,156,109)(48,191,145,116)>;
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), (1,25,7,31)(2,36,8,30)(3,35,9,29)(4,34,10,28)(5,33,11,27)(6,32,12,26)(13,192,19,186)(14,191,20,185)(15,190,21,184)(16,189,22,183)(17,188,23,182)(18,187,24,181)(37,53,43,59)(38,52,44,58)(39,51,45,57)(40,50,46,56)(41,49,47,55)(42,60,48,54)(61,174,67,180)(62,173,68,179)(63,172,69,178)(64,171,70,177)(65,170,71,176)(66,169,72,175)(73,138,79,144)(74,137,80,143)(75,136,81,142)(76,135,82,141)(77,134,83,140)(78,133,84,139)(85,101,91,107)(86,100,92,106)(87,99,93,105)(88,98,94,104)(89,97,95,103)(90,108,96,102)(109,167,115,161)(110,166,116,160)(111,165,117,159)(112,164,118,158)(113,163,119,157)(114,162,120,168)(121,154,127,148)(122,153,128,147)(123,152,129,146)(124,151,130,145)(125,150,131,156)(126,149,132,155), (1,182,74,119)(2,187,75,112)(3,192,76,117)(4,185,77,110)(5,190,78,115)(6,183,79,120)(7,188,80,113)(8,181,81,118)(9,186,82,111)(10,191,83,116)(11,184,84,109)(12,189,73,114)(13,141,165,29)(14,134,166,34)(15,139,167,27)(16,144,168,32)(17,137,157,25)(18,142,158,30)(19,135,159,35)(20,140,160,28)(21,133,161,33)(22,138,162,26)(23,143,163,31)(24,136,164,36)(37,170,146,104)(38,175,147,97)(39,180,148,102)(40,173,149,107)(41,178,150,100)(42,171,151,105)(43,176,152,98)(44,169,153,103)(45,174,154,108)(46,179,155,101)(47,172,156,106)(48,177,145,99)(49,63,131,92)(50,68,132,85)(51,61,121,90)(52,66,122,95)(53,71,123,88)(54,64,124,93)(55,69,125,86)(56,62,126,91)(57,67,127,96)(58,72,128,89)(59,65,129,94)(60,70,130,87), (1,108,74,174)(2,103,75,169)(3,98,76,176)(4,105,77,171)(5,100,78,178)(6,107,79,173)(7,102,80,180)(8,97,81,175)(9,104,82,170)(10,99,83,177)(11,106,84,172)(12,101,73,179)(13,132,165,50)(14,127,166,57)(15,122,167,52)(16,129,168,59)(17,124,157,54)(18,131,158,49)(19,126,159,56)(20,121,160,51)(21,128,161,58)(22,123,162,53)(23,130,163,60)(24,125,164,55)(25,93,137,64)(26,88,138,71)(27,95,139,66)(28,90,140,61)(29,85,141,68)(30,92,142,63)(31,87,143,70)(32,94,144,65)(33,89,133,72)(34,96,134,67)(35,91,135,62)(36,86,136,69)(37,186,146,111)(38,181,147,118)(39,188,148,113)(40,183,149,120)(41,190,150,115)(42,185,151,110)(43,192,152,117)(44,187,153,112)(45,182,154,119)(46,189,155,114)(47,184,156,109)(48,191,145,116) );
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)], [(1,25,7,31),(2,36,8,30),(3,35,9,29),(4,34,10,28),(5,33,11,27),(6,32,12,26),(13,192,19,186),(14,191,20,185),(15,190,21,184),(16,189,22,183),(17,188,23,182),(18,187,24,181),(37,53,43,59),(38,52,44,58),(39,51,45,57),(40,50,46,56),(41,49,47,55),(42,60,48,54),(61,174,67,180),(62,173,68,179),(63,172,69,178),(64,171,70,177),(65,170,71,176),(66,169,72,175),(73,138,79,144),(74,137,80,143),(75,136,81,142),(76,135,82,141),(77,134,83,140),(78,133,84,139),(85,101,91,107),(86,100,92,106),(87,99,93,105),(88,98,94,104),(89,97,95,103),(90,108,96,102),(109,167,115,161),(110,166,116,160),(111,165,117,159),(112,164,118,158),(113,163,119,157),(114,162,120,168),(121,154,127,148),(122,153,128,147),(123,152,129,146),(124,151,130,145),(125,150,131,156),(126,149,132,155)], [(1,182,74,119),(2,187,75,112),(3,192,76,117),(4,185,77,110),(5,190,78,115),(6,183,79,120),(7,188,80,113),(8,181,81,118),(9,186,82,111),(10,191,83,116),(11,184,84,109),(12,189,73,114),(13,141,165,29),(14,134,166,34),(15,139,167,27),(16,144,168,32),(17,137,157,25),(18,142,158,30),(19,135,159,35),(20,140,160,28),(21,133,161,33),(22,138,162,26),(23,143,163,31),(24,136,164,36),(37,170,146,104),(38,175,147,97),(39,180,148,102),(40,173,149,107),(41,178,150,100),(42,171,151,105),(43,176,152,98),(44,169,153,103),(45,174,154,108),(46,179,155,101),(47,172,156,106),(48,177,145,99),(49,63,131,92),(50,68,132,85),(51,61,121,90),(52,66,122,95),(53,71,123,88),(54,64,124,93),(55,69,125,86),(56,62,126,91),(57,67,127,96),(58,72,128,89),(59,65,129,94),(60,70,130,87)], [(1,108,74,174),(2,103,75,169),(3,98,76,176),(4,105,77,171),(5,100,78,178),(6,107,79,173),(7,102,80,180),(8,97,81,175),(9,104,82,170),(10,99,83,177),(11,106,84,172),(12,101,73,179),(13,132,165,50),(14,127,166,57),(15,122,167,52),(16,129,168,59),(17,124,157,54),(18,131,158,49),(19,126,159,56),(20,121,160,51),(21,128,161,58),(22,123,162,53),(23,130,163,60),(24,125,164,55),(25,93,137,64),(26,88,138,71),(27,95,139,66),(28,90,140,61),(29,85,141,68),(30,92,142,63),(31,87,143,70),(32,94,144,65),(33,89,133,72),(34,96,134,67),(35,91,135,62),(36,86,136,69),(37,186,146,111),(38,181,147,118),(39,188,148,113),(40,183,149,120),(41,190,150,115),(42,185,151,110),(43,192,152,117),(44,187,153,112),(45,182,154,119),(46,189,155,114),(47,184,156,109),(48,191,145,116)]])
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 | SD16 | C4○D4 | C4○D12 | C8.C22 | S3×Q8 | S3×D4 | S3×SD16 | D4.D6 |
| kernel | Dic6⋊Q8 | C12.Q8 | C6.SD16 | Dic3⋊C8 | C2.Dic12 | C3×C4.Q8 | Dic6⋊C4 | C12⋊Q8 | C4.Q8 | Dic6 | C2×Dic3 | C4⋊C4 | C2×C8 | Dic3 | C12 | C4 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 2 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of Dic6⋊Q8 ►in GL4(𝔽73) generated by
| 0 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 1 | 1 |
| 21 | 54 | 0 | 0 |
| 54 | 52 | 0 | 0 |
| 0 | 0 | 45 | 3 |
| 0 | 0 | 31 | 28 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 47 | 8 |
| 0 | 0 | 34 | 26 |
| 12 | 52 | 0 | 0 |
| 52 | 61 | 0 | 0 |
| 0 | 0 | 7 | 14 |
| 0 | 0 | 59 | 66 |
G:=sub<GL(4,GF(73))| [0,72,0,0,1,0,0,0,0,0,0,1,0,0,72,1],[21,54,0,0,54,52,0,0,0,0,45,31,0,0,3,28],[72,0,0,0,0,72,0,0,0,0,47,34,0,0,8,26],[12,52,0,0,52,61,0,0,0,0,7,59,0,0,14,66] >;
Dic6⋊Q8 in GAP, Magma, Sage, TeX
{\rm Dic}_6\rtimes Q_8 % in TeX
G:=Group("Dic6:Q8"); // GroupNames label
G:=SmallGroup(192,413);
// by ID
G=gap.SmallGroup(192,413);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,120,135,268,570,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=c^4=1,b^2=a^6,d^2=c^2,b*a*b^-1=a^-1,c*a*c^-1=a^5,d*a*d^-1=a^7,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations