metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6⋊4Q8, C42.42D6, C12.14SD16, C4⋊C8.10S3, C4.47(S3×Q8), C3⋊2(Q8⋊Q8), (C2×C8).134D6, C8⋊Dic3.9C2, C4.8(C24⋊C2), (C2×C4).138D12, (C2×C12).127D4, C12.106(C2×Q8), C6.13(C2×SD16), (C4×C12).77C22, C12⋊2Q8.12C2, (C4×Dic6).13C2, C6.34(C22⋊Q8), C2.Dic12.5C2, C12.290(C4○D4), (C2×C12).761C23, (C2×C24).141C22, C2.15(C4.D12), C2.21(C8.D6), C22.124(C2×D12), C6.18(C8.C22), C4⋊Dic3.22C22, C4.114(D4⋊2S3), (C2×Dic6).217C22, (C3×C4⋊C8).15C2, C2.16(C2×C24⋊C2), (C2×C6).144(C2×D4), (C2×C4).706(C22×S3), SmallGroup(192,410)
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=cac-1=a-1, ad=da, cbc-1=a9b, bd=db, dcd-1=c-1 >
Subgroups: 264 in 96 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C3, C4, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Dic3, C12, C12, C12, C2×C6, C42, C42, C4⋊C4, C2×C8, C2×Q8, C24, Dic6, Dic6, C2×Dic3, C2×C12, Q8⋊C4, C4⋊C8, C4.Q8, C4×Q8, C4⋊Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C4⋊Dic3, C4×C12, C2×C24, C2×Dic6, C2×Dic6, Q8⋊Q8, C2.Dic12, C8⋊Dic3, C3×C4⋊C8, C4×Dic6, C12⋊2Q8, Dic6⋊4Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, SD16, C2×D4, C2×Q8, C4○D4, D12, C22×S3, C22⋊Q8, C2×SD16, C8.C22, C24⋊C2, C2×D12, D4⋊2S3, S3×Q8, Q8⋊Q8, C4.D12, C2×C24⋊C2, C8.D6, 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)
(1 109 7 115)(2 120 8 114)(3 119 9 113)(4 118 10 112)(5 117 11 111)(6 116 12 110)(13 33 19 27)(14 32 20 26)(15 31 21 25)(16 30 22 36)(17 29 23 35)(18 28 24 34)(37 125 43 131)(38 124 44 130)(39 123 45 129)(40 122 46 128)(41 121 47 127)(42 132 48 126)(49 142 55 136)(50 141 56 135)(51 140 57 134)(52 139 58 133)(53 138 59 144)(54 137 60 143)(61 171 67 177)(62 170 68 176)(63 169 69 175)(64 180 70 174)(65 179 71 173)(66 178 72 172)(73 190 79 184)(74 189 80 183)(75 188 81 182)(76 187 82 181)(77 186 83 192)(78 185 84 191)(85 104 91 98)(86 103 92 97)(87 102 93 108)(88 101 94 107)(89 100 95 106)(90 99 96 105)(145 159 151 165)(146 158 152 164)(147 157 153 163)(148 168 154 162)(149 167 155 161)(150 166 156 160)
(1 44 186 69)(2 43 187 68)(3 42 188 67)(4 41 189 66)(5 40 190 65)(6 39 191 64)(7 38 192 63)(8 37 181 62)(9 48 182 61)(10 47 183 72)(11 46 184 71)(12 45 185 70)(13 149 144 106)(14 148 133 105)(15 147 134 104)(16 146 135 103)(17 145 136 102)(18 156 137 101)(19 155 138 100)(20 154 139 99)(21 153 140 98)(22 152 141 97)(23 151 142 108)(24 150 143 107)(25 166 57 88)(26 165 58 87)(27 164 59 86)(28 163 60 85)(29 162 49 96)(30 161 50 95)(31 160 51 94)(32 159 52 93)(33 158 53 92)(34 157 54 91)(35 168 55 90)(36 167 56 89)(73 176 111 131)(74 175 112 130)(75 174 113 129)(76 173 114 128)(77 172 115 127)(78 171 116 126)(79 170 117 125)(80 169 118 124)(81 180 119 123)(82 179 120 122)(83 178 109 121)(84 177 110 132)
(1 149 186 106)(2 150 187 107)(3 151 188 108)(4 152 189 97)(5 153 190 98)(6 154 191 99)(7 155 192 100)(8 156 181 101)(9 145 182 102)(10 146 183 103)(11 147 184 104)(12 148 185 105)(13 69 144 44)(14 70 133 45)(15 71 134 46)(16 72 135 47)(17 61 136 48)(18 62 137 37)(19 63 138 38)(20 64 139 39)(21 65 140 40)(22 66 141 41)(23 67 142 42)(24 68 143 43)(25 179 57 122)(26 180 58 123)(27 169 59 124)(28 170 60 125)(29 171 49 126)(30 172 50 127)(31 173 51 128)(32 174 52 129)(33 175 53 130)(34 176 54 131)(35 177 55 132)(36 178 56 121)(73 91 111 157)(74 92 112 158)(75 93 113 159)(76 94 114 160)(77 95 115 161)(78 96 116 162)(79 85 117 163)(80 86 118 164)(81 87 119 165)(82 88 120 166)(83 89 109 167)(84 90 110 168)
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), (1,109,7,115)(2,120,8,114)(3,119,9,113)(4,118,10,112)(5,117,11,111)(6,116,12,110)(13,33,19,27)(14,32,20,26)(15,31,21,25)(16,30,22,36)(17,29,23,35)(18,28,24,34)(37,125,43,131)(38,124,44,130)(39,123,45,129)(40,122,46,128)(41,121,47,127)(42,132,48,126)(49,142,55,136)(50,141,56,135)(51,140,57,134)(52,139,58,133)(53,138,59,144)(54,137,60,143)(61,171,67,177)(62,170,68,176)(63,169,69,175)(64,180,70,174)(65,179,71,173)(66,178,72,172)(73,190,79,184)(74,189,80,183)(75,188,81,182)(76,187,82,181)(77,186,83,192)(78,185,84,191)(85,104,91,98)(86,103,92,97)(87,102,93,108)(88,101,94,107)(89,100,95,106)(90,99,96,105)(145,159,151,165)(146,158,152,164)(147,157,153,163)(148,168,154,162)(149,167,155,161)(150,166,156,160), (1,44,186,69)(2,43,187,68)(3,42,188,67)(4,41,189,66)(5,40,190,65)(6,39,191,64)(7,38,192,63)(8,37,181,62)(9,48,182,61)(10,47,183,72)(11,46,184,71)(12,45,185,70)(13,149,144,106)(14,148,133,105)(15,147,134,104)(16,146,135,103)(17,145,136,102)(18,156,137,101)(19,155,138,100)(20,154,139,99)(21,153,140,98)(22,152,141,97)(23,151,142,108)(24,150,143,107)(25,166,57,88)(26,165,58,87)(27,164,59,86)(28,163,60,85)(29,162,49,96)(30,161,50,95)(31,160,51,94)(32,159,52,93)(33,158,53,92)(34,157,54,91)(35,168,55,90)(36,167,56,89)(73,176,111,131)(74,175,112,130)(75,174,113,129)(76,173,114,128)(77,172,115,127)(78,171,116,126)(79,170,117,125)(80,169,118,124)(81,180,119,123)(82,179,120,122)(83,178,109,121)(84,177,110,132), (1,149,186,106)(2,150,187,107)(3,151,188,108)(4,152,189,97)(5,153,190,98)(6,154,191,99)(7,155,192,100)(8,156,181,101)(9,145,182,102)(10,146,183,103)(11,147,184,104)(12,148,185,105)(13,69,144,44)(14,70,133,45)(15,71,134,46)(16,72,135,47)(17,61,136,48)(18,62,137,37)(19,63,138,38)(20,64,139,39)(21,65,140,40)(22,66,141,41)(23,67,142,42)(24,68,143,43)(25,179,57,122)(26,180,58,123)(27,169,59,124)(28,170,60,125)(29,171,49,126)(30,172,50,127)(31,173,51,128)(32,174,52,129)(33,175,53,130)(34,176,54,131)(35,177,55,132)(36,178,56,121)(73,91,111,157)(74,92,112,158)(75,93,113,159)(76,94,114,160)(77,95,115,161)(78,96,116,162)(79,85,117,163)(80,86,118,164)(81,87,119,165)(82,88,120,166)(83,89,109,167)(84,90,110,168)>;
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), (1,109,7,115)(2,120,8,114)(3,119,9,113)(4,118,10,112)(5,117,11,111)(6,116,12,110)(13,33,19,27)(14,32,20,26)(15,31,21,25)(16,30,22,36)(17,29,23,35)(18,28,24,34)(37,125,43,131)(38,124,44,130)(39,123,45,129)(40,122,46,128)(41,121,47,127)(42,132,48,126)(49,142,55,136)(50,141,56,135)(51,140,57,134)(52,139,58,133)(53,138,59,144)(54,137,60,143)(61,171,67,177)(62,170,68,176)(63,169,69,175)(64,180,70,174)(65,179,71,173)(66,178,72,172)(73,190,79,184)(74,189,80,183)(75,188,81,182)(76,187,82,181)(77,186,83,192)(78,185,84,191)(85,104,91,98)(86,103,92,97)(87,102,93,108)(88,101,94,107)(89,100,95,106)(90,99,96,105)(145,159,151,165)(146,158,152,164)(147,157,153,163)(148,168,154,162)(149,167,155,161)(150,166,156,160), (1,44,186,69)(2,43,187,68)(3,42,188,67)(4,41,189,66)(5,40,190,65)(6,39,191,64)(7,38,192,63)(8,37,181,62)(9,48,182,61)(10,47,183,72)(11,46,184,71)(12,45,185,70)(13,149,144,106)(14,148,133,105)(15,147,134,104)(16,146,135,103)(17,145,136,102)(18,156,137,101)(19,155,138,100)(20,154,139,99)(21,153,140,98)(22,152,141,97)(23,151,142,108)(24,150,143,107)(25,166,57,88)(26,165,58,87)(27,164,59,86)(28,163,60,85)(29,162,49,96)(30,161,50,95)(31,160,51,94)(32,159,52,93)(33,158,53,92)(34,157,54,91)(35,168,55,90)(36,167,56,89)(73,176,111,131)(74,175,112,130)(75,174,113,129)(76,173,114,128)(77,172,115,127)(78,171,116,126)(79,170,117,125)(80,169,118,124)(81,180,119,123)(82,179,120,122)(83,178,109,121)(84,177,110,132), (1,149,186,106)(2,150,187,107)(3,151,188,108)(4,152,189,97)(5,153,190,98)(6,154,191,99)(7,155,192,100)(8,156,181,101)(9,145,182,102)(10,146,183,103)(11,147,184,104)(12,148,185,105)(13,69,144,44)(14,70,133,45)(15,71,134,46)(16,72,135,47)(17,61,136,48)(18,62,137,37)(19,63,138,38)(20,64,139,39)(21,65,140,40)(22,66,141,41)(23,67,142,42)(24,68,143,43)(25,179,57,122)(26,180,58,123)(27,169,59,124)(28,170,60,125)(29,171,49,126)(30,172,50,127)(31,173,51,128)(32,174,52,129)(33,175,53,130)(34,176,54,131)(35,177,55,132)(36,178,56,121)(73,91,111,157)(74,92,112,158)(75,93,113,159)(76,94,114,160)(77,95,115,161)(78,96,116,162)(79,85,117,163)(80,86,118,164)(81,87,119,165)(82,88,120,166)(83,89,109,167)(84,90,110,168) );
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)], [(1,109,7,115),(2,120,8,114),(3,119,9,113),(4,118,10,112),(5,117,11,111),(6,116,12,110),(13,33,19,27),(14,32,20,26),(15,31,21,25),(16,30,22,36),(17,29,23,35),(18,28,24,34),(37,125,43,131),(38,124,44,130),(39,123,45,129),(40,122,46,128),(41,121,47,127),(42,132,48,126),(49,142,55,136),(50,141,56,135),(51,140,57,134),(52,139,58,133),(53,138,59,144),(54,137,60,143),(61,171,67,177),(62,170,68,176),(63,169,69,175),(64,180,70,174),(65,179,71,173),(66,178,72,172),(73,190,79,184),(74,189,80,183),(75,188,81,182),(76,187,82,181),(77,186,83,192),(78,185,84,191),(85,104,91,98),(86,103,92,97),(87,102,93,108),(88,101,94,107),(89,100,95,106),(90,99,96,105),(145,159,151,165),(146,158,152,164),(147,157,153,163),(148,168,154,162),(149,167,155,161),(150,166,156,160)], [(1,44,186,69),(2,43,187,68),(3,42,188,67),(4,41,189,66),(5,40,190,65),(6,39,191,64),(7,38,192,63),(8,37,181,62),(9,48,182,61),(10,47,183,72),(11,46,184,71),(12,45,185,70),(13,149,144,106),(14,148,133,105),(15,147,134,104),(16,146,135,103),(17,145,136,102),(18,156,137,101),(19,155,138,100),(20,154,139,99),(21,153,140,98),(22,152,141,97),(23,151,142,108),(24,150,143,107),(25,166,57,88),(26,165,58,87),(27,164,59,86),(28,163,60,85),(29,162,49,96),(30,161,50,95),(31,160,51,94),(32,159,52,93),(33,158,53,92),(34,157,54,91),(35,168,55,90),(36,167,56,89),(73,176,111,131),(74,175,112,130),(75,174,113,129),(76,173,114,128),(77,172,115,127),(78,171,116,126),(79,170,117,125),(80,169,118,124),(81,180,119,123),(82,179,120,122),(83,178,109,121),(84,177,110,132)], [(1,149,186,106),(2,150,187,107),(3,151,188,108),(4,152,189,97),(5,153,190,98),(6,154,191,99),(7,155,192,100),(8,156,181,101),(9,145,182,102),(10,146,183,103),(11,147,184,104),(12,148,185,105),(13,69,144,44),(14,70,133,45),(15,71,134,46),(16,72,135,47),(17,61,136,48),(18,62,137,37),(19,63,138,38),(20,64,139,39),(21,65,140,40),(22,66,141,41),(23,67,142,42),(24,68,143,43),(25,179,57,122),(26,180,58,123),(27,169,59,124),(28,170,60,125),(29,171,49,126),(30,172,50,127),(31,173,51,128),(32,174,52,129),(33,175,53,130),(34,176,54,131),(35,177,55,132),(36,178,56,121),(73,91,111,157),(74,92,112,158),(75,93,113,159),(76,94,114,160),(77,95,115,161),(78,96,116,162),(79,85,117,163),(80,86,118,164),(81,87,119,165),(82,88,120,166),(83,89,109,167),(84,90,110,168)]])
39 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 | 12G | 12H | 24A | ··· | 24H |
| 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 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 12 | 12 | 12 | 12 | 24 | 24 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | - | - | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D4 | D6 | D6 | SD16 | C4○D4 | D12 | C24⋊C2 | C8.C22 | D4⋊2S3 | S3×Q8 | C8.D6 |
| kernel | Dic6⋊4Q8 | C2.Dic12 | C8⋊Dic3 | C3×C4⋊C8 | C4×Dic6 | C12⋊2Q8 | C4⋊C8 | Dic6 | C2×C12 | C42 | C2×C8 | C12 | C12 | C2×C4 | C4 | C6 | C4 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 4 | 2 | 4 | 8 | 1 | 1 | 1 | 2 |
Matrix representation of Dic6⋊4Q8 ►in GL6(𝔽73)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 54 | 67 |
| 0 | 0 | 0 | 0 | 36 | 19 |
| 51 | 10 | 0 | 0 | 0 | 0 |
| 32 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 42 | 3 | 0 | 0 |
| 0 | 0 | 45 | 31 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 41 |
| 0 | 0 | 0 | 0 | 41 | 56 |
| 51 | 10 | 0 | 0 | 0 | 0 |
| 32 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 23 | 0 | 0 |
| 0 | 0 | 28 | 42 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 3 | 61 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 65 | 0 | 0 |
| 0 | 0 | 26 | 39 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,54,36,0,0,0,0,67,19],[51,32,0,0,0,0,10,22,0,0,0,0,0,0,42,45,0,0,0,0,3,31,0,0,0,0,0,0,17,41,0,0,0,0,41,56],[51,32,0,0,0,0,10,22,0,0,0,0,0,0,31,28,0,0,0,0,23,42,0,0,0,0,0,0,12,3,0,0,0,0,1,61],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,34,26,0,0,0,0,65,39,0,0,0,0,0,0,72,0,0,0,0,0,0,72] >;
Dic6⋊4Q8 in GAP, Magma, Sage, TeX
{\rm Dic}_6\rtimes_4Q_8 % in TeX
G:=Group("Dic6:4Q8"); // GroupNames label
G:=SmallGroup(192,410);
// by ID
G=gap.SmallGroup(192,410);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,120,254,219,142,1123,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=c^4=1,b^2=a^6,d^2=c^2,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^9*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations