metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6.4Q8, C42.67D6, C4.9(S3×Q8), C4⋊C4.72D6, C3⋊7(Q8.Q8), C12.31(C2×Q8), (C2×C12).273D4, C12⋊C8.20C2, C42.C2.2S3, C6.107(C4○D8), C12.69(C4○D4), (C4×Dic6).16C2, C6.Q16.12C2, C6.73(C22⋊Q8), (C4×C12).111C22, (C2×C12).381C23, C2.10(D6⋊3Q8), C4.32(Q8⋊3S3), C6.SD16.12C2, C12.Q8.13C2, C2.26(Q8.13D6), C2.20(Q8.14D6), C6.121(C8.C22), C4⋊Dic3.342C22, (C2×Dic6).273C22, (C2×C6).512(C2×D4), (C2×C4).64(C3⋊D4), (C2×C3⋊C8).124C22, (C3×C42.C2).1C2, (C3×C4⋊C4).119C22, (C2×C4).479(C22×S3), C22.185(C2×C3⋊D4), SmallGroup(192,622)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic6.4Q8
G = < a,b,c,d | a12=c4=1, b2=a6, d2=c2, bab-1=dad-1=a-1, cac-1=a7, cbc-1=a9b, bd=db, dcd-1=a3c-1 >
Subgroups: 208 in 90 conjugacy classes, 41 normal (39 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Dic3, C12, C12, C2×C6, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C3⋊C8, Dic6, Dic6, C2×Dic3, C2×C12, C2×C12, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×Q8, C42.C2, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, Q8.Q8, C12⋊C8, C6.Q16, C12.Q8, C6.SD16, C4×Dic6, C3×C42.C2, Dic6.4Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, C3⋊D4, C22×S3, C22⋊Q8, C4○D8, C8.C22, S3×Q8, Q8⋊3S3, C2×C3⋊D4, Q8.Q8, D6⋊3Q8, Q8.13D6, Q8.14D6, Dic6.4Q8
►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 80 7 74)(2 79 8 73)(3 78 9 84)(4 77 10 83)(5 76 11 82)(6 75 12 81)(13 95 19 89)(14 94 20 88)(15 93 21 87)(16 92 22 86)(17 91 23 85)(18 90 24 96)(25 40 31 46)(26 39 32 45)(27 38 33 44)(28 37 34 43)(29 48 35 42)(30 47 36 41)(49 101 55 107)(50 100 56 106)(51 99 57 105)(52 98 58 104)(53 97 59 103)(54 108 60 102)(61 161 67 167)(62 160 68 166)(63 159 69 165)(64 158 70 164)(65 157 71 163)(66 168 72 162)(109 142 115 136)(110 141 116 135)(111 140 117 134)(112 139 118 133)(113 138 119 144)(114 137 120 143)(121 174 127 180)(122 173 128 179)(123 172 129 178)(124 171 130 177)(125 170 131 176)(126 169 132 175)(145 192 151 186)(146 191 152 185)(147 190 153 184)(148 189 154 183)(149 188 155 182)(150 187 156 181)
(1 38 88 158)(2 45 89 165)(3 40 90 160)(4 47 91 167)(5 42 92 162)(6 37 93 157)(7 44 94 164)(8 39 95 159)(9 46 96 166)(10 41 85 161)(11 48 86 168)(12 43 87 163)(13 66 79 29)(14 61 80 36)(15 68 81 31)(16 63 82 26)(17 70 83 33)(18 65 84 28)(19 72 73 35)(20 67 74 30)(21 62 75 25)(22 69 76 32)(23 64 77 27)(24 71 78 34)(49 120 131 151)(50 115 132 146)(51 110 121 153)(52 117 122 148)(53 112 123 155)(54 119 124 150)(55 114 125 145)(56 109 126 152)(57 116 127 147)(58 111 128 154)(59 118 129 149)(60 113 130 156)(97 142 172 185)(98 137 173 192)(99 144 174 187)(100 139 175 182)(101 134 176 189)(102 141 177 184)(103 136 178 191)(104 143 179 186)(105 138 180 181)(106 133 169 188)(107 140 170 183)(108 135 171 190)
(1 148 88 117)(2 147 89 116)(3 146 90 115)(4 145 91 114)(5 156 92 113)(6 155 93 112)(7 154 94 111)(8 153 95 110)(9 152 96 109)(10 151 85 120)(11 150 86 119)(12 149 87 118)(13 135 79 190)(14 134 80 189)(15 133 81 188)(16 144 82 187)(17 143 83 186)(18 142 84 185)(19 141 73 184)(20 140 74 183)(21 139 75 182)(22 138 76 181)(23 137 77 192)(24 136 78 191)(25 172 62 97)(26 171 63 108)(27 170 64 107)(28 169 65 106)(29 180 66 105)(30 179 67 104)(31 178 68 103)(32 177 69 102)(33 176 70 101)(34 175 71 100)(35 174 72 99)(36 173 61 98)(37 132 157 50)(38 131 158 49)(39 130 159 60)(40 129 160 59)(41 128 161 58)(42 127 162 57)(43 126 163 56)(44 125 164 55)(45 124 165 54)(46 123 166 53)(47 122 167 52)(48 121 168 51)
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), (1,80,7,74)(2,79,8,73)(3,78,9,84)(4,77,10,83)(5,76,11,82)(6,75,12,81)(13,95,19,89)(14,94,20,88)(15,93,21,87)(16,92,22,86)(17,91,23,85)(18,90,24,96)(25,40,31,46)(26,39,32,45)(27,38,33,44)(28,37,34,43)(29,48,35,42)(30,47,36,41)(49,101,55,107)(50,100,56,106)(51,99,57,105)(52,98,58,104)(53,97,59,103)(54,108,60,102)(61,161,67,167)(62,160,68,166)(63,159,69,165)(64,158,70,164)(65,157,71,163)(66,168,72,162)(109,142,115,136)(110,141,116,135)(111,140,117,134)(112,139,118,133)(113,138,119,144)(114,137,120,143)(121,174,127,180)(122,173,128,179)(123,172,129,178)(124,171,130,177)(125,170,131,176)(126,169,132,175)(145,192,151,186)(146,191,152,185)(147,190,153,184)(148,189,154,183)(149,188,155,182)(150,187,156,181), (1,38,88,158)(2,45,89,165)(3,40,90,160)(4,47,91,167)(5,42,92,162)(6,37,93,157)(7,44,94,164)(8,39,95,159)(9,46,96,166)(10,41,85,161)(11,48,86,168)(12,43,87,163)(13,66,79,29)(14,61,80,36)(15,68,81,31)(16,63,82,26)(17,70,83,33)(18,65,84,28)(19,72,73,35)(20,67,74,30)(21,62,75,25)(22,69,76,32)(23,64,77,27)(24,71,78,34)(49,120,131,151)(50,115,132,146)(51,110,121,153)(52,117,122,148)(53,112,123,155)(54,119,124,150)(55,114,125,145)(56,109,126,152)(57,116,127,147)(58,111,128,154)(59,118,129,149)(60,113,130,156)(97,142,172,185)(98,137,173,192)(99,144,174,187)(100,139,175,182)(101,134,176,189)(102,141,177,184)(103,136,178,191)(104,143,179,186)(105,138,180,181)(106,133,169,188)(107,140,170,183)(108,135,171,190), (1,148,88,117)(2,147,89,116)(3,146,90,115)(4,145,91,114)(5,156,92,113)(6,155,93,112)(7,154,94,111)(8,153,95,110)(9,152,96,109)(10,151,85,120)(11,150,86,119)(12,149,87,118)(13,135,79,190)(14,134,80,189)(15,133,81,188)(16,144,82,187)(17,143,83,186)(18,142,84,185)(19,141,73,184)(20,140,74,183)(21,139,75,182)(22,138,76,181)(23,137,77,192)(24,136,78,191)(25,172,62,97)(26,171,63,108)(27,170,64,107)(28,169,65,106)(29,180,66,105)(30,179,67,104)(31,178,68,103)(32,177,69,102)(33,176,70,101)(34,175,71,100)(35,174,72,99)(36,173,61,98)(37,132,157,50)(38,131,158,49)(39,130,159,60)(40,129,160,59)(41,128,161,58)(42,127,162,57)(43,126,163,56)(44,125,164,55)(45,124,165,54)(46,123,166,53)(47,122,167,52)(48,121,168,51)>;
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), (1,80,7,74)(2,79,8,73)(3,78,9,84)(4,77,10,83)(5,76,11,82)(6,75,12,81)(13,95,19,89)(14,94,20,88)(15,93,21,87)(16,92,22,86)(17,91,23,85)(18,90,24,96)(25,40,31,46)(26,39,32,45)(27,38,33,44)(28,37,34,43)(29,48,35,42)(30,47,36,41)(49,101,55,107)(50,100,56,106)(51,99,57,105)(52,98,58,104)(53,97,59,103)(54,108,60,102)(61,161,67,167)(62,160,68,166)(63,159,69,165)(64,158,70,164)(65,157,71,163)(66,168,72,162)(109,142,115,136)(110,141,116,135)(111,140,117,134)(112,139,118,133)(113,138,119,144)(114,137,120,143)(121,174,127,180)(122,173,128,179)(123,172,129,178)(124,171,130,177)(125,170,131,176)(126,169,132,175)(145,192,151,186)(146,191,152,185)(147,190,153,184)(148,189,154,183)(149,188,155,182)(150,187,156,181), (1,38,88,158)(2,45,89,165)(3,40,90,160)(4,47,91,167)(5,42,92,162)(6,37,93,157)(7,44,94,164)(8,39,95,159)(9,46,96,166)(10,41,85,161)(11,48,86,168)(12,43,87,163)(13,66,79,29)(14,61,80,36)(15,68,81,31)(16,63,82,26)(17,70,83,33)(18,65,84,28)(19,72,73,35)(20,67,74,30)(21,62,75,25)(22,69,76,32)(23,64,77,27)(24,71,78,34)(49,120,131,151)(50,115,132,146)(51,110,121,153)(52,117,122,148)(53,112,123,155)(54,119,124,150)(55,114,125,145)(56,109,126,152)(57,116,127,147)(58,111,128,154)(59,118,129,149)(60,113,130,156)(97,142,172,185)(98,137,173,192)(99,144,174,187)(100,139,175,182)(101,134,176,189)(102,141,177,184)(103,136,178,191)(104,143,179,186)(105,138,180,181)(106,133,169,188)(107,140,170,183)(108,135,171,190), (1,148,88,117)(2,147,89,116)(3,146,90,115)(4,145,91,114)(5,156,92,113)(6,155,93,112)(7,154,94,111)(8,153,95,110)(9,152,96,109)(10,151,85,120)(11,150,86,119)(12,149,87,118)(13,135,79,190)(14,134,80,189)(15,133,81,188)(16,144,82,187)(17,143,83,186)(18,142,84,185)(19,141,73,184)(20,140,74,183)(21,139,75,182)(22,138,76,181)(23,137,77,192)(24,136,78,191)(25,172,62,97)(26,171,63,108)(27,170,64,107)(28,169,65,106)(29,180,66,105)(30,179,67,104)(31,178,68,103)(32,177,69,102)(33,176,70,101)(34,175,71,100)(35,174,72,99)(36,173,61,98)(37,132,157,50)(38,131,158,49)(39,130,159,60)(40,129,160,59)(41,128,161,58)(42,127,162,57)(43,126,163,56)(44,125,164,55)(45,124,165,54)(46,123,166,53)(47,122,167,52)(48,121,168,51) );
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)], [(1,80,7,74),(2,79,8,73),(3,78,9,84),(4,77,10,83),(5,76,11,82),(6,75,12,81),(13,95,19,89),(14,94,20,88),(15,93,21,87),(16,92,22,86),(17,91,23,85),(18,90,24,96),(25,40,31,46),(26,39,32,45),(27,38,33,44),(28,37,34,43),(29,48,35,42),(30,47,36,41),(49,101,55,107),(50,100,56,106),(51,99,57,105),(52,98,58,104),(53,97,59,103),(54,108,60,102),(61,161,67,167),(62,160,68,166),(63,159,69,165),(64,158,70,164),(65,157,71,163),(66,168,72,162),(109,142,115,136),(110,141,116,135),(111,140,117,134),(112,139,118,133),(113,138,119,144),(114,137,120,143),(121,174,127,180),(122,173,128,179),(123,172,129,178),(124,171,130,177),(125,170,131,176),(126,169,132,175),(145,192,151,186),(146,191,152,185),(147,190,153,184),(148,189,154,183),(149,188,155,182),(150,187,156,181)], [(1,38,88,158),(2,45,89,165),(3,40,90,160),(4,47,91,167),(5,42,92,162),(6,37,93,157),(7,44,94,164),(8,39,95,159),(9,46,96,166),(10,41,85,161),(11,48,86,168),(12,43,87,163),(13,66,79,29),(14,61,80,36),(15,68,81,31),(16,63,82,26),(17,70,83,33),(18,65,84,28),(19,72,73,35),(20,67,74,30),(21,62,75,25),(22,69,76,32),(23,64,77,27),(24,71,78,34),(49,120,131,151),(50,115,132,146),(51,110,121,153),(52,117,122,148),(53,112,123,155),(54,119,124,150),(55,114,125,145),(56,109,126,152),(57,116,127,147),(58,111,128,154),(59,118,129,149),(60,113,130,156),(97,142,172,185),(98,137,173,192),(99,144,174,187),(100,139,175,182),(101,134,176,189),(102,141,177,184),(103,136,178,191),(104,143,179,186),(105,138,180,181),(106,133,169,188),(107,140,170,183),(108,135,171,190)], [(1,148,88,117),(2,147,89,116),(3,146,90,115),(4,145,91,114),(5,156,92,113),(6,155,93,112),(7,154,94,111),(8,153,95,110),(9,152,96,109),(10,151,85,120),(11,150,86,119),(12,149,87,118),(13,135,79,190),(14,134,80,189),(15,133,81,188),(16,144,82,187),(17,143,83,186),(18,142,84,185),(19,141,73,184),(20,140,74,183),(21,139,75,182),(22,138,76,181),(23,137,77,192),(24,136,78,191),(25,172,62,97),(26,171,63,108),(27,170,64,107),(28,169,65,106),(29,180,66,105),(30,179,67,104),(31,178,68,103),(32,177,69,102),(33,176,70,101),(34,175,71,100),(35,174,72,99),(36,173,61,98),(37,132,157,50),(38,131,158,49),(39,130,159,60),(40,129,160,59),(41,128,161,58),(42,127,162,57),(43,126,163,56),(44,125,164,55),(45,124,165,54),(46,123,166,53),(47,122,167,52),(48,121,168,51)]])
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 | ··· | 12F | 12G | 12H | 12I | 12J |
| 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 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
33 irreducible representations
| dim | 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 | S3 | Q8 | D4 | D6 | D6 | C4○D4 | C3⋊D4 | C4○D8 | C8.C22 | S3×Q8 | Q8⋊3S3 | Q8.13D6 | Q8.14D6 |
| kernel | Dic6.4Q8 | C12⋊C8 | C6.Q16 | C12.Q8 | C6.SD16 | C4×Dic6 | C3×C42.C2 | C42.C2 | Dic6 | C2×C12 | C42 | C4⋊C4 | C12 | C2×C4 | C6 | C6 | C4 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of Dic6.4Q8 ►in GL6(𝔽73)
| 1 | 48 | 0 | 0 | 0 | 0 |
| 3 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 69 | 38 | 0 | 0 | 0 | 0 |
| 13 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 5 | 0 | 0 |
| 0 | 0 | 23 | 55 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 21 | 57 | 0 | 0 | 0 | 0 |
| 64 | 52 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 38 | 4 | 0 | 0 | 0 | 0 |
| 59 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 5 | 0 | 0 |
| 0 | 0 | 23 | 55 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 1 | 61 |
G:=sub<GL(6,GF(73))| [1,3,0,0,0,0,48,72,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[69,13,0,0,0,0,38,4,0,0,0,0,0,0,18,23,0,0,0,0,5,55,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[21,64,0,0,0,0,57,52,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[38,59,0,0,0,0,4,35,0,0,0,0,0,0,18,23,0,0,0,0,5,55,0,0,0,0,0,0,12,1,0,0,0,0,1,61] >;
Dic6.4Q8 in GAP, Magma, Sage, TeX
{\rm Dic}_6._4Q_8 % in TeX
G:=Group("Dic6.4Q8"); // GroupNames label
G:=SmallGroup(192,622);
// by ID
G=gap.SmallGroup(192,622);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,344,254,219,100,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=c^4=1,b^2=a^6,d^2=c^2,b*a*b^-1=d*a*d^-1=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^9*b,b*d=d*b,d*c*d^-1=a^3*c^-1>;
// generators/relations