metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6⋊5Q8, C12.10Q16, C42.84D6, C4⋊C4.86D6, C4⋊Q8.10S3, C4.13(S3×Q8), C3⋊5(C4.Q16), C6.43(C2×Q16), C12.40(C2×Q8), (C2×C12).160D4, C12⋊C8.22C2, C4.8(C3⋊Q16), C12.86(C4○D4), (C4×Dic6).18C2, C6.Q16.17C2, C6.77(C22⋊Q8), C6.100(C8⋊C22), (C4×C12).138C22, (C2×C12).409C23, C4.36(Q8⋊3S3), C2.14(D6⋊3Q8), C6.SD16.14C2, C2.21(D12⋊6C22), C4⋊Dic3.350C22, (C2×Dic6).276C22, (C3×C4⋊Q8).10C2, (C2×C6).540(C2×D4), C2.14(C2×C3⋊Q16), (C2×C3⋊C8).141C22, (C2×C4).191(C3⋊D4), (C3×C4⋊C4).133C22, (C2×C4).506(C22×S3), C22.212(C2×C3⋊D4), SmallGroup(192,650)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic6⋊5Q8
G = < a,b,c,d | a12=c4=1, b2=a6, d2=c2, bab-1=a-1, cac-1=a7, ad=da, cbc-1=a9b, dbd-1=a6b, dcd-1=c-1 >
Subgroups: 224 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, C4⋊C4, C2×C8, C2×Q8, C3⋊C8, Dic6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, Q8⋊C4, C4⋊C8, C2.D8, C4×Q8, C4⋊Q8, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C6×Q8, C4.Q16, C12⋊C8, C6.Q16, C6.SD16, C4×Dic6, C3×C4⋊Q8, Dic6⋊5Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, Q16, C2×D4, C2×Q8, C4○D4, C3⋊D4, C22×S3, C22⋊Q8, C2×Q16, C8⋊C22, C3⋊Q16, S3×Q8, Q8⋊3S3, C2×C3⋊D4, C4.Q16, D12⋊6C22, C2×C3⋊Q16, D6⋊3Q8, Dic6⋊5Q8
►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 43 7 37)(2 42 8 48)(3 41 9 47)(4 40 10 46)(5 39 11 45)(6 38 12 44)(13 119 19 113)(14 118 20 112)(15 117 21 111)(16 116 22 110)(17 115 23 109)(18 114 24 120)(25 94 31 88)(26 93 32 87)(27 92 33 86)(28 91 34 85)(29 90 35 96)(30 89 36 95)(49 68 55 62)(50 67 56 61)(51 66 57 72)(52 65 58 71)(53 64 59 70)(54 63 60 69)(73 184 79 190)(74 183 80 189)(75 182 81 188)(76 181 82 187)(77 192 83 186)(78 191 84 185)(97 131 103 125)(98 130 104 124)(99 129 105 123)(100 128 106 122)(101 127 107 121)(102 126 108 132)(133 159 139 165)(134 158 140 164)(135 157 141 163)(136 168 142 162)(137 167 143 161)(138 166 144 160)(145 177 151 171)(146 176 152 170)(147 175 153 169)(148 174 154 180)(149 173 155 179)(150 172 156 178)
(1 133 23 121)(2 140 24 128)(3 135 13 123)(4 142 14 130)(5 137 15 125)(6 144 16 132)(7 139 17 127)(8 134 18 122)(9 141 19 129)(10 136 20 124)(11 143 21 131)(12 138 22 126)(25 58 74 156)(26 53 75 151)(27 60 76 146)(28 55 77 153)(29 50 78 148)(30 57 79 155)(31 52 80 150)(32 59 81 145)(33 54 82 152)(34 49 83 147)(35 56 84 154)(36 51 73 149)(37 168 115 98)(38 163 116 105)(39 158 117 100)(40 165 118 107)(41 160 119 102)(42 167 120 97)(43 162 109 104)(44 157 110 99)(45 164 111 106)(46 159 112 101)(47 166 113 108)(48 161 114 103)(61 188 180 87)(62 183 169 94)(63 190 170 89)(64 185 171 96)(65 192 172 91)(66 187 173 86)(67 182 174 93)(68 189 175 88)(69 184 176 95)(70 191 177 90)(71 186 178 85)(72 181 179 92)
(1 93 23 182)(2 94 24 183)(3 95 13 184)(4 96 14 185)(5 85 15 186)(6 86 16 187)(7 87 17 188)(8 88 18 189)(9 89 19 190)(10 90 20 191)(11 91 21 192)(12 92 22 181)(25 120 74 42)(26 109 75 43)(27 110 76 44)(28 111 77 45)(29 112 78 46)(30 113 79 47)(31 114 80 48)(32 115 81 37)(33 116 82 38)(34 117 83 39)(35 118 84 40)(36 119 73 41)(49 158 147 100)(50 159 148 101)(51 160 149 102)(52 161 150 103)(53 162 151 104)(54 163 152 105)(55 164 153 106)(56 165 154 107)(57 166 155 108)(58 167 156 97)(59 168 145 98)(60 157 146 99)(61 139 180 127)(62 140 169 128)(63 141 170 129)(64 142 171 130)(65 143 172 131)(66 144 173 132)(67 133 174 121)(68 134 175 122)(69 135 176 123)(70 136 177 124)(71 137 178 125)(72 138 179 126)
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), (1,43,7,37)(2,42,8,48)(3,41,9,47)(4,40,10,46)(5,39,11,45)(6,38,12,44)(13,119,19,113)(14,118,20,112)(15,117,21,111)(16,116,22,110)(17,115,23,109)(18,114,24,120)(25,94,31,88)(26,93,32,87)(27,92,33,86)(28,91,34,85)(29,90,35,96)(30,89,36,95)(49,68,55,62)(50,67,56,61)(51,66,57,72)(52,65,58,71)(53,64,59,70)(54,63,60,69)(73,184,79,190)(74,183,80,189)(75,182,81,188)(76,181,82,187)(77,192,83,186)(78,191,84,185)(97,131,103,125)(98,130,104,124)(99,129,105,123)(100,128,106,122)(101,127,107,121)(102,126,108,132)(133,159,139,165)(134,158,140,164)(135,157,141,163)(136,168,142,162)(137,167,143,161)(138,166,144,160)(145,177,151,171)(146,176,152,170)(147,175,153,169)(148,174,154,180)(149,173,155,179)(150,172,156,178), (1,133,23,121)(2,140,24,128)(3,135,13,123)(4,142,14,130)(5,137,15,125)(6,144,16,132)(7,139,17,127)(8,134,18,122)(9,141,19,129)(10,136,20,124)(11,143,21,131)(12,138,22,126)(25,58,74,156)(26,53,75,151)(27,60,76,146)(28,55,77,153)(29,50,78,148)(30,57,79,155)(31,52,80,150)(32,59,81,145)(33,54,82,152)(34,49,83,147)(35,56,84,154)(36,51,73,149)(37,168,115,98)(38,163,116,105)(39,158,117,100)(40,165,118,107)(41,160,119,102)(42,167,120,97)(43,162,109,104)(44,157,110,99)(45,164,111,106)(46,159,112,101)(47,166,113,108)(48,161,114,103)(61,188,180,87)(62,183,169,94)(63,190,170,89)(64,185,171,96)(65,192,172,91)(66,187,173,86)(67,182,174,93)(68,189,175,88)(69,184,176,95)(70,191,177,90)(71,186,178,85)(72,181,179,92), (1,93,23,182)(2,94,24,183)(3,95,13,184)(4,96,14,185)(5,85,15,186)(6,86,16,187)(7,87,17,188)(8,88,18,189)(9,89,19,190)(10,90,20,191)(11,91,21,192)(12,92,22,181)(25,120,74,42)(26,109,75,43)(27,110,76,44)(28,111,77,45)(29,112,78,46)(30,113,79,47)(31,114,80,48)(32,115,81,37)(33,116,82,38)(34,117,83,39)(35,118,84,40)(36,119,73,41)(49,158,147,100)(50,159,148,101)(51,160,149,102)(52,161,150,103)(53,162,151,104)(54,163,152,105)(55,164,153,106)(56,165,154,107)(57,166,155,108)(58,167,156,97)(59,168,145,98)(60,157,146,99)(61,139,180,127)(62,140,169,128)(63,141,170,129)(64,142,171,130)(65,143,172,131)(66,144,173,132)(67,133,174,121)(68,134,175,122)(69,135,176,123)(70,136,177,124)(71,137,178,125)(72,138,179,126)>;
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), (1,43,7,37)(2,42,8,48)(3,41,9,47)(4,40,10,46)(5,39,11,45)(6,38,12,44)(13,119,19,113)(14,118,20,112)(15,117,21,111)(16,116,22,110)(17,115,23,109)(18,114,24,120)(25,94,31,88)(26,93,32,87)(27,92,33,86)(28,91,34,85)(29,90,35,96)(30,89,36,95)(49,68,55,62)(50,67,56,61)(51,66,57,72)(52,65,58,71)(53,64,59,70)(54,63,60,69)(73,184,79,190)(74,183,80,189)(75,182,81,188)(76,181,82,187)(77,192,83,186)(78,191,84,185)(97,131,103,125)(98,130,104,124)(99,129,105,123)(100,128,106,122)(101,127,107,121)(102,126,108,132)(133,159,139,165)(134,158,140,164)(135,157,141,163)(136,168,142,162)(137,167,143,161)(138,166,144,160)(145,177,151,171)(146,176,152,170)(147,175,153,169)(148,174,154,180)(149,173,155,179)(150,172,156,178), (1,133,23,121)(2,140,24,128)(3,135,13,123)(4,142,14,130)(5,137,15,125)(6,144,16,132)(7,139,17,127)(8,134,18,122)(9,141,19,129)(10,136,20,124)(11,143,21,131)(12,138,22,126)(25,58,74,156)(26,53,75,151)(27,60,76,146)(28,55,77,153)(29,50,78,148)(30,57,79,155)(31,52,80,150)(32,59,81,145)(33,54,82,152)(34,49,83,147)(35,56,84,154)(36,51,73,149)(37,168,115,98)(38,163,116,105)(39,158,117,100)(40,165,118,107)(41,160,119,102)(42,167,120,97)(43,162,109,104)(44,157,110,99)(45,164,111,106)(46,159,112,101)(47,166,113,108)(48,161,114,103)(61,188,180,87)(62,183,169,94)(63,190,170,89)(64,185,171,96)(65,192,172,91)(66,187,173,86)(67,182,174,93)(68,189,175,88)(69,184,176,95)(70,191,177,90)(71,186,178,85)(72,181,179,92), (1,93,23,182)(2,94,24,183)(3,95,13,184)(4,96,14,185)(5,85,15,186)(6,86,16,187)(7,87,17,188)(8,88,18,189)(9,89,19,190)(10,90,20,191)(11,91,21,192)(12,92,22,181)(25,120,74,42)(26,109,75,43)(27,110,76,44)(28,111,77,45)(29,112,78,46)(30,113,79,47)(31,114,80,48)(32,115,81,37)(33,116,82,38)(34,117,83,39)(35,118,84,40)(36,119,73,41)(49,158,147,100)(50,159,148,101)(51,160,149,102)(52,161,150,103)(53,162,151,104)(54,163,152,105)(55,164,153,106)(56,165,154,107)(57,166,155,108)(58,167,156,97)(59,168,145,98)(60,157,146,99)(61,139,180,127)(62,140,169,128)(63,141,170,129)(64,142,171,130)(65,143,172,131)(66,144,173,132)(67,133,174,121)(68,134,175,122)(69,135,176,123)(70,136,177,124)(71,137,178,125)(72,138,179,126) );
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)], [(1,43,7,37),(2,42,8,48),(3,41,9,47),(4,40,10,46),(5,39,11,45),(6,38,12,44),(13,119,19,113),(14,118,20,112),(15,117,21,111),(16,116,22,110),(17,115,23,109),(18,114,24,120),(25,94,31,88),(26,93,32,87),(27,92,33,86),(28,91,34,85),(29,90,35,96),(30,89,36,95),(49,68,55,62),(50,67,56,61),(51,66,57,72),(52,65,58,71),(53,64,59,70),(54,63,60,69),(73,184,79,190),(74,183,80,189),(75,182,81,188),(76,181,82,187),(77,192,83,186),(78,191,84,185),(97,131,103,125),(98,130,104,124),(99,129,105,123),(100,128,106,122),(101,127,107,121),(102,126,108,132),(133,159,139,165),(134,158,140,164),(135,157,141,163),(136,168,142,162),(137,167,143,161),(138,166,144,160),(145,177,151,171),(146,176,152,170),(147,175,153,169),(148,174,154,180),(149,173,155,179),(150,172,156,178)], [(1,133,23,121),(2,140,24,128),(3,135,13,123),(4,142,14,130),(5,137,15,125),(6,144,16,132),(7,139,17,127),(8,134,18,122),(9,141,19,129),(10,136,20,124),(11,143,21,131),(12,138,22,126),(25,58,74,156),(26,53,75,151),(27,60,76,146),(28,55,77,153),(29,50,78,148),(30,57,79,155),(31,52,80,150),(32,59,81,145),(33,54,82,152),(34,49,83,147),(35,56,84,154),(36,51,73,149),(37,168,115,98),(38,163,116,105),(39,158,117,100),(40,165,118,107),(41,160,119,102),(42,167,120,97),(43,162,109,104),(44,157,110,99),(45,164,111,106),(46,159,112,101),(47,166,113,108),(48,161,114,103),(61,188,180,87),(62,183,169,94),(63,190,170,89),(64,185,171,96),(65,192,172,91),(66,187,173,86),(67,182,174,93),(68,189,175,88),(69,184,176,95),(70,191,177,90),(71,186,178,85),(72,181,179,92)], [(1,93,23,182),(2,94,24,183),(3,95,13,184),(4,96,14,185),(5,85,15,186),(6,86,16,187),(7,87,17,188),(8,88,18,189),(9,89,19,190),(10,90,20,191),(11,91,21,192),(12,92,22,181),(25,120,74,42),(26,109,75,43),(27,110,76,44),(28,111,77,45),(29,112,78,46),(30,113,79,47),(31,114,80,48),(32,115,81,37),(33,116,82,38),(34,117,83,39),(35,118,84,40),(36,119,73,41),(49,158,147,100),(50,159,148,101),(51,160,149,102),(52,161,150,103),(53,162,151,104),(54,163,152,105),(55,164,153,106),(56,165,154,107),(57,166,155,108),(58,167,156,97),(59,168,145,98),(60,157,146,99),(61,139,180,127),(62,140,169,128),(63,141,170,129),(64,142,171,130),(65,143,172,131),(66,144,173,132),(67,133,174,121),(68,134,175,122),(69,135,176,123),(70,136,177,124),(71,137,178,125),(72,138,179,126)]])
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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | - | + | - | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D4 | D6 | D6 | Q16 | C4○D4 | C3⋊D4 | C8⋊C22 | C3⋊Q16 | S3×Q8 | Q8⋊3S3 | D12⋊6C22 |
| kernel | Dic6⋊5Q8 | C12⋊C8 | C6.Q16 | C6.SD16 | C4×Dic6 | C3×C4⋊Q8 | C4⋊Q8 | Dic6 | C2×C12 | C42 | C4⋊C4 | C12 | C12 | C2×C4 | C6 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 4 | 2 | 4 | 1 | 2 | 1 | 1 | 2 |
Matrix representation of Dic6⋊5Q8 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 46 | 19 | 0 | 0 | 0 | 0 |
| 27 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 42 | 3 | 0 | 0 |
| 0 | 0 | 45 | 31 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 21 |
| 0 | 0 | 0 | 0 | 21 | 54 |
| 27 | 0 | 0 | 0 | 0 | 0 |
| 46 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 43 | 60 | 0 | 0 |
| 0 | 0 | 13 | 30 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 17 |
| 0 | 0 | 0 | 0 | 17 | 41 |
| 1 | 2 | 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 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[46,27,0,0,0,0,19,27,0,0,0,0,0,0,42,45,0,0,0,0,3,31,0,0,0,0,0,0,19,21,0,0,0,0,21,54],[27,46,0,0,0,0,0,46,0,0,0,0,0,0,43,13,0,0,0,0,60,30,0,0,0,0,0,0,32,17,0,0,0,0,17,41],[1,72,0,0,0,0,2,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0] >;
Dic6⋊5Q8 in GAP, Magma, Sage, TeX
{\rm Dic}_6\rtimes_5Q_8 % in TeX
G:=Group("Dic6:5Q8"); // GroupNames label
G:=SmallGroup(192,650);
// by ID
G=gap.SmallGroup(192,650);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,120,254,219,268,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=c^4=1,b^2=a^6,d^2=c^2,b*a*b^-1=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=a^9*b,d*b*d^-1=a^6*b,d*c*d^-1=c^-1>;
// generators/relations