metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6⋊10Q8, C42.121D6, C6.642- (1+4), C4.49(S3×Q8), C4⋊C4.321D6, C3⋊1(Q8⋊3Q8), (C4×Q8).17S3, (Q8×C12).11C2, (C2×Q8).199D6, C12.107(C2×Q8), C4.17(C4○D12), C6.27(C22×Q8), (C2×C6).111C24, C2.21(Q8○D12), C12⋊2Q8.24C2, Dic3.10(C2×Q8), (C4×Dic6).21C2, Dic3.Q8.1C2, C12.115(C4○D4), (C2×C12).589C23, (C4×C12).164C22, Dic3⋊Q8.7C2, C4⋊Dic3.42C22, C12.6Q8.10C2, (C6×Q8).211C22, Dic6⋊C4.10C2, C22.136(S3×C23), (C4×Dic3).80C22, (C2×Dic3).50C23, Dic3⋊C4.114C22, (C2×Dic6).146C22, C2.11(C2×S3×Q8), C6.52(C2×C4○D4), C2.59(C2×C4○D12), (C3×C4⋊C4).339C22, (C2×C4).166(C22×S3), SmallGroup(192,1126)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 392 in 200 conjugacy classes, 107 normal (29 characteristic)
C1, C2 [×3], C3, C4 [×4], C4 [×15], C22, C6 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×8], Q8 [×10], Dic3 [×4], Dic3 [×6], C12 [×4], C12 [×5], C2×C6, C42, C42 [×2], C42 [×6], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×19], C2×Q8, C2×Q8 [×3], Dic6 [×4], Dic6 [×4], C2×Dic3 [×8], C2×C12 [×3], C2×C12 [×4], C3×Q8 [×2], C4×Q8, C4×Q8 [×5], C42.C2 [×6], C4⋊Q8 [×3], C4×Dic3 [×6], Dic3⋊C4 [×14], C4⋊Dic3, C4⋊Dic3 [×4], C4×C12, C4×C12 [×2], C3×C4⋊C4, C3×C4⋊C4 [×2], C2×Dic6, C2×Dic6 [×2], C6×Q8, Q8⋊3Q8, C4×Dic6, C4×Dic6 [×2], C12⋊2Q8, C12.6Q8 [×2], Dic6⋊C4 [×2], Dic3.Q8 [×4], Dic3⋊Q8 [×2], Q8×C12, Dic6⋊10Q8
Quotients:
C1, C2 [×15], C22 [×35], S3, Q8 [×4], C23 [×15], D6 [×7], C2×Q8 [×6], C4○D4 [×2], C24, C22×S3 [×7], C22×Q8, C2×C4○D4, 2- (1+4), C4○D12 [×2], S3×Q8 [×2], S3×C23, Q8⋊3Q8, C2×C4○D12, C2×S3×Q8, Q8○D12, Dic6⋊10Q8
Generators and relations
G = < a,b,c,d | a12=c4=1, b2=a6, d2=c2, bab-1=a-1, ac=ca, ad=da, cbc-1=a6b, bd=db, dcd-1=c-1 >
►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)
(1 148 7 154)(2 147 8 153)(3 146 9 152)(4 145 10 151)(5 156 11 150)(6 155 12 149)(13 103 19 97)(14 102 20 108)(15 101 21 107)(16 100 22 106)(17 99 23 105)(18 98 24 104)(25 75 31 81)(26 74 32 80)(27 73 33 79)(28 84 34 78)(29 83 35 77)(30 82 36 76)(37 137 43 143)(38 136 44 142)(39 135 45 141)(40 134 46 140)(41 133 47 139)(42 144 48 138)(49 90 55 96)(50 89 56 95)(51 88 57 94)(52 87 58 93)(53 86 59 92)(54 85 60 91)(61 183 67 189)(62 182 68 188)(63 181 69 187)(64 192 70 186)(65 191 71 185)(66 190 72 184)(109 159 115 165)(110 158 116 164)(111 157 117 163)(112 168 118 162)(113 167 119 161)(114 166 120 160)(121 173 127 179)(122 172 128 178)(123 171 129 177)(124 170 130 176)(125 169 131 175)(126 180 132 174)
(1 25 157 99)(2 26 158 100)(3 27 159 101)(4 28 160 102)(5 29 161 103)(6 30 162 104)(7 31 163 105)(8 32 164 106)(9 33 165 107)(10 34 166 108)(11 35 167 97)(12 36 168 98)(13 156 77 113)(14 145 78 114)(15 146 79 115)(16 147 80 116)(17 148 81 117)(18 149 82 118)(19 150 83 119)(20 151 84 120)(21 152 73 109)(22 153 74 110)(23 154 75 111)(24 155 76 112)(37 122 90 186)(38 123 91 187)(39 124 92 188)(40 125 93 189)(41 126 94 190)(42 127 95 191)(43 128 96 192)(44 129 85 181)(45 130 86 182)(46 131 87 183)(47 132 88 184)(48 121 89 185)(49 64 143 172)(50 65 144 173)(51 66 133 174)(52 67 134 175)(53 68 135 176)(54 69 136 177)(55 70 137 178)(56 71 138 179)(57 72 139 180)(58 61 140 169)(59 62 141 170)(60 63 142 171)
(1 121 157 185)(2 122 158 186)(3 123 159 187)(4 124 160 188)(5 125 161 189)(6 126 162 190)(7 127 163 191)(8 128 164 192)(9 129 165 181)(10 130 166 182)(11 131 167 183)(12 132 168 184)(13 58 77 140)(14 59 78 141)(15 60 79 142)(16 49 80 143)(17 50 81 144)(18 51 82 133)(19 52 83 134)(20 53 84 135)(21 54 73 136)(22 55 74 137)(23 56 75 138)(24 57 76 139)(25 48 99 89)(26 37 100 90)(27 38 101 91)(28 39 102 92)(29 40 103 93)(30 41 104 94)(31 42 105 95)(32 43 106 96)(33 44 107 85)(34 45 108 86)(35 46 97 87)(36 47 98 88)(61 156 169 113)(62 145 170 114)(63 146 171 115)(64 147 172 116)(65 148 173 117)(66 149 174 118)(67 150 175 119)(68 151 176 120)(69 152 177 109)(70 153 178 110)(71 154 179 111)(72 155 180 112)
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), (1,148,7,154)(2,147,8,153)(3,146,9,152)(4,145,10,151)(5,156,11,150)(6,155,12,149)(13,103,19,97)(14,102,20,108)(15,101,21,107)(16,100,22,106)(17,99,23,105)(18,98,24,104)(25,75,31,81)(26,74,32,80)(27,73,33,79)(28,84,34,78)(29,83,35,77)(30,82,36,76)(37,137,43,143)(38,136,44,142)(39,135,45,141)(40,134,46,140)(41,133,47,139)(42,144,48,138)(49,90,55,96)(50,89,56,95)(51,88,57,94)(52,87,58,93)(53,86,59,92)(54,85,60,91)(61,183,67,189)(62,182,68,188)(63,181,69,187)(64,192,70,186)(65,191,71,185)(66,190,72,184)(109,159,115,165)(110,158,116,164)(111,157,117,163)(112,168,118,162)(113,167,119,161)(114,166,120,160)(121,173,127,179)(122,172,128,178)(123,171,129,177)(124,170,130,176)(125,169,131,175)(126,180,132,174), (1,25,157,99)(2,26,158,100)(3,27,159,101)(4,28,160,102)(5,29,161,103)(6,30,162,104)(7,31,163,105)(8,32,164,106)(9,33,165,107)(10,34,166,108)(11,35,167,97)(12,36,168,98)(13,156,77,113)(14,145,78,114)(15,146,79,115)(16,147,80,116)(17,148,81,117)(18,149,82,118)(19,150,83,119)(20,151,84,120)(21,152,73,109)(22,153,74,110)(23,154,75,111)(24,155,76,112)(37,122,90,186)(38,123,91,187)(39,124,92,188)(40,125,93,189)(41,126,94,190)(42,127,95,191)(43,128,96,192)(44,129,85,181)(45,130,86,182)(46,131,87,183)(47,132,88,184)(48,121,89,185)(49,64,143,172)(50,65,144,173)(51,66,133,174)(52,67,134,175)(53,68,135,176)(54,69,136,177)(55,70,137,178)(56,71,138,179)(57,72,139,180)(58,61,140,169)(59,62,141,170)(60,63,142,171), (1,121,157,185)(2,122,158,186)(3,123,159,187)(4,124,160,188)(5,125,161,189)(6,126,162,190)(7,127,163,191)(8,128,164,192)(9,129,165,181)(10,130,166,182)(11,131,167,183)(12,132,168,184)(13,58,77,140)(14,59,78,141)(15,60,79,142)(16,49,80,143)(17,50,81,144)(18,51,82,133)(19,52,83,134)(20,53,84,135)(21,54,73,136)(22,55,74,137)(23,56,75,138)(24,57,76,139)(25,48,99,89)(26,37,100,90)(27,38,101,91)(28,39,102,92)(29,40,103,93)(30,41,104,94)(31,42,105,95)(32,43,106,96)(33,44,107,85)(34,45,108,86)(35,46,97,87)(36,47,98,88)(61,156,169,113)(62,145,170,114)(63,146,171,115)(64,147,172,116)(65,148,173,117)(66,149,174,118)(67,150,175,119)(68,151,176,120)(69,152,177,109)(70,153,178,110)(71,154,179,111)(72,155,180,112)>;
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), (1,148,7,154)(2,147,8,153)(3,146,9,152)(4,145,10,151)(5,156,11,150)(6,155,12,149)(13,103,19,97)(14,102,20,108)(15,101,21,107)(16,100,22,106)(17,99,23,105)(18,98,24,104)(25,75,31,81)(26,74,32,80)(27,73,33,79)(28,84,34,78)(29,83,35,77)(30,82,36,76)(37,137,43,143)(38,136,44,142)(39,135,45,141)(40,134,46,140)(41,133,47,139)(42,144,48,138)(49,90,55,96)(50,89,56,95)(51,88,57,94)(52,87,58,93)(53,86,59,92)(54,85,60,91)(61,183,67,189)(62,182,68,188)(63,181,69,187)(64,192,70,186)(65,191,71,185)(66,190,72,184)(109,159,115,165)(110,158,116,164)(111,157,117,163)(112,168,118,162)(113,167,119,161)(114,166,120,160)(121,173,127,179)(122,172,128,178)(123,171,129,177)(124,170,130,176)(125,169,131,175)(126,180,132,174), (1,25,157,99)(2,26,158,100)(3,27,159,101)(4,28,160,102)(5,29,161,103)(6,30,162,104)(7,31,163,105)(8,32,164,106)(9,33,165,107)(10,34,166,108)(11,35,167,97)(12,36,168,98)(13,156,77,113)(14,145,78,114)(15,146,79,115)(16,147,80,116)(17,148,81,117)(18,149,82,118)(19,150,83,119)(20,151,84,120)(21,152,73,109)(22,153,74,110)(23,154,75,111)(24,155,76,112)(37,122,90,186)(38,123,91,187)(39,124,92,188)(40,125,93,189)(41,126,94,190)(42,127,95,191)(43,128,96,192)(44,129,85,181)(45,130,86,182)(46,131,87,183)(47,132,88,184)(48,121,89,185)(49,64,143,172)(50,65,144,173)(51,66,133,174)(52,67,134,175)(53,68,135,176)(54,69,136,177)(55,70,137,178)(56,71,138,179)(57,72,139,180)(58,61,140,169)(59,62,141,170)(60,63,142,171), (1,121,157,185)(2,122,158,186)(3,123,159,187)(4,124,160,188)(5,125,161,189)(6,126,162,190)(7,127,163,191)(8,128,164,192)(9,129,165,181)(10,130,166,182)(11,131,167,183)(12,132,168,184)(13,58,77,140)(14,59,78,141)(15,60,79,142)(16,49,80,143)(17,50,81,144)(18,51,82,133)(19,52,83,134)(20,53,84,135)(21,54,73,136)(22,55,74,137)(23,56,75,138)(24,57,76,139)(25,48,99,89)(26,37,100,90)(27,38,101,91)(28,39,102,92)(29,40,103,93)(30,41,104,94)(31,42,105,95)(32,43,106,96)(33,44,107,85)(34,45,108,86)(35,46,97,87)(36,47,98,88)(61,156,169,113)(62,145,170,114)(63,146,171,115)(64,147,172,116)(65,148,173,117)(66,149,174,118)(67,150,175,119)(68,151,176,120)(69,152,177,109)(70,153,178,110)(71,154,179,111)(72,155,180,112) );
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)], [(1,148,7,154),(2,147,8,153),(3,146,9,152),(4,145,10,151),(5,156,11,150),(6,155,12,149),(13,103,19,97),(14,102,20,108),(15,101,21,107),(16,100,22,106),(17,99,23,105),(18,98,24,104),(25,75,31,81),(26,74,32,80),(27,73,33,79),(28,84,34,78),(29,83,35,77),(30,82,36,76),(37,137,43,143),(38,136,44,142),(39,135,45,141),(40,134,46,140),(41,133,47,139),(42,144,48,138),(49,90,55,96),(50,89,56,95),(51,88,57,94),(52,87,58,93),(53,86,59,92),(54,85,60,91),(61,183,67,189),(62,182,68,188),(63,181,69,187),(64,192,70,186),(65,191,71,185),(66,190,72,184),(109,159,115,165),(110,158,116,164),(111,157,117,163),(112,168,118,162),(113,167,119,161),(114,166,120,160),(121,173,127,179),(122,172,128,178),(123,171,129,177),(124,170,130,176),(125,169,131,175),(126,180,132,174)], [(1,25,157,99),(2,26,158,100),(3,27,159,101),(4,28,160,102),(5,29,161,103),(6,30,162,104),(7,31,163,105),(8,32,164,106),(9,33,165,107),(10,34,166,108),(11,35,167,97),(12,36,168,98),(13,156,77,113),(14,145,78,114),(15,146,79,115),(16,147,80,116),(17,148,81,117),(18,149,82,118),(19,150,83,119),(20,151,84,120),(21,152,73,109),(22,153,74,110),(23,154,75,111),(24,155,76,112),(37,122,90,186),(38,123,91,187),(39,124,92,188),(40,125,93,189),(41,126,94,190),(42,127,95,191),(43,128,96,192),(44,129,85,181),(45,130,86,182),(46,131,87,183),(47,132,88,184),(48,121,89,185),(49,64,143,172),(50,65,144,173),(51,66,133,174),(52,67,134,175),(53,68,135,176),(54,69,136,177),(55,70,137,178),(56,71,138,179),(57,72,139,180),(58,61,140,169),(59,62,141,170),(60,63,142,171)], [(1,121,157,185),(2,122,158,186),(3,123,159,187),(4,124,160,188),(5,125,161,189),(6,126,162,190),(7,127,163,191),(8,128,164,192),(9,129,165,181),(10,130,166,182),(11,131,167,183),(12,132,168,184),(13,58,77,140),(14,59,78,141),(15,60,79,142),(16,49,80,143),(17,50,81,144),(18,51,82,133),(19,52,83,134),(20,53,84,135),(21,54,73,136),(22,55,74,137),(23,56,75,138),(24,57,76,139),(25,48,99,89),(26,37,100,90),(27,38,101,91),(28,39,102,92),(29,40,103,93),(30,41,104,94),(31,42,105,95),(32,43,106,96),(33,44,107,85),(34,45,108,86),(35,46,97,87),(36,47,98,88),(61,156,169,113),(62,145,170,114),(63,146,171,115),(64,147,172,116),(65,148,173,117),(66,149,174,118),(67,150,175,119),(68,151,176,120),(69,152,177,109),(70,153,178,110),(71,154,179,111),(72,155,180,112)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 7 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 2 | 0 | 0 |
| 6 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 10 | 4 |
| 0 | 0 | 4 | 3 |
G:=sub<GL(4,GF(13))| [7,0,0,0,0,2,0,0,0,0,1,0,0,0,0,1],[0,6,0,0,2,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,12,0,0,0,0,0,12,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,10,4,0,0,4,3] >;
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | ··· | 4U | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | - | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | D6 | C4○D4 | C4○D12 | 2- (1+4) | S3×Q8 | Q8○D12 |
| kernel | Dic6⋊10Q8 | C4×Dic6 | C12⋊2Q8 | C12.6Q8 | Dic6⋊C4 | Dic3.Q8 | Dic3⋊Q8 | Q8×C12 | C4×Q8 | Dic6 | C42 | C4⋊C4 | C2×Q8 | C12 | C4 | C6 | C4 | C2 |
| # reps | 1 | 3 | 1 | 2 | 2 | 4 | 2 | 1 | 1 | 4 | 3 | 3 | 1 | 4 | 8 | 1 | 2 | 2 |
In GAP, Magma, Sage, TeX
Dic_6\rtimes_{10}Q_8 % in TeX
G:=Group("Dic6:10Q8"); // GroupNames label
G:=SmallGroup(192,1126);
// by ID
G=gap.SmallGroup(192,1126);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,477,232,100,185,192,6278]);
// 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,a*c=c*a,a*d=d*a,c*b*c^-1=a^6*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀