direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8×Dic6, C42.120D6, C6.1082+ (1+4), C3⋊1(Q82), (C3×Q8)⋊5Q8, C4.48(S3×Q8), C4⋊C4.290D6, C12⋊Q8.10C2, (C4×Q8).16S3, C12.44(C2×Q8), (Q8×C12).10C2, (C2×Q8).220D6, Dic3.9(C2×Q8), C4.17(C2×Dic6), C2.21(D4○D12), C6.15(C22×Q8), (C2×C6).110C24, C12⋊2Q8.23C2, (Q8×Dic3).10C2, (C4×Dic6).20C2, (C2×C12).167C23, (C4×C12).163C22, (C6×Q8).210C22, C2.17(C22×Dic6), C4⋊Dic3.201C22, C22.135(S3×C23), (C2×Dic6).30C22, (C4×Dic3).79C22, Dic3⋊C4.113C22, (C2×Dic3).210C23, C2.10(C2×S3×Q8), (C3×C4⋊C4).338C22, (C2×C4).582(C22×S3), SmallGroup(192,1125)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 456 in 212 conjugacy classes, 123 normal (18 characteristic)
C1, C2 [×3], C3, C4 [×8], C4 [×13], C22, C6 [×3], C2×C4, C2×C4 [×6], C2×C4 [×8], Q8 [×4], Q8 [×10], Dic3 [×4], Dic3 [×6], C12 [×8], C12 [×3], C2×C6, C42 [×3], C42 [×6], C4⋊C4 [×3], C4⋊C4 [×15], C2×Q8, C2×Q8 [×7], Dic6 [×4], Dic6 [×6], C2×Dic3 [×8], C2×C12, C2×C12 [×6], C3×Q8 [×4], C4×Q8, C4×Q8 [×5], C4⋊Q8 [×9], C4×Dic3 [×6], Dic3⋊C4 [×6], C4⋊Dic3 [×9], C4×C12 [×3], C3×C4⋊C4 [×3], C2×Dic6, C2×Dic6 [×6], C6×Q8, Q82, C4×Dic6 [×3], C12⋊2Q8 [×3], C12⋊Q8 [×6], Q8×Dic3 [×2], Q8×C12, Q8×Dic6
Quotients:
C1, C2 [×15], C22 [×35], S3, Q8 [×8], C23 [×15], D6 [×7], C2×Q8 [×12], C24, Dic6 [×4], C22×S3 [×7], C22×Q8 [×2], 2+ (1+4), C2×Dic6 [×6], S3×Q8 [×2], S3×C23, Q82, C22×Dic6, C2×S3×Q8, D4○D12, Q8×Dic6
Generators and relations
G = < a,b,c,d | a4=c12=1, b2=a2, d2=c6, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
►Regular action on 192 points
Generators in S192
(1 43 180 151)(2 44 169 152)(3 45 170 153)(4 46 171 154)(5 47 172 155)(6 48 173 156)(7 37 174 145)(8 38 175 146)(9 39 176 147)(10 40 177 148)(11 41 178 149)(12 42 179 150)(13 61 27 90)(14 62 28 91)(15 63 29 92)(16 64 30 93)(17 65 31 94)(18 66 32 95)(19 67 33 96)(20 68 34 85)(21 69 35 86)(22 70 36 87)(23 71 25 88)(24 72 26 89)(49 122 81 160)(50 123 82 161)(51 124 83 162)(52 125 84 163)(53 126 73 164)(54 127 74 165)(55 128 75 166)(56 129 76 167)(57 130 77 168)(58 131 78 157)(59 132 79 158)(60 121 80 159)(97 114 140 185)(98 115 141 186)(99 116 142 187)(100 117 143 188)(101 118 144 189)(102 119 133 190)(103 120 134 191)(104 109 135 192)(105 110 136 181)(106 111 137 182)(107 112 138 183)(108 113 139 184)
(1 58 180 78)(2 59 169 79)(3 60 170 80)(4 49 171 81)(5 50 172 82)(6 51 173 83)(7 52 174 84)(8 53 175 73)(9 54 176 74)(10 55 177 75)(11 56 178 76)(12 57 179 77)(13 134 27 103)(14 135 28 104)(15 136 29 105)(16 137 30 106)(17 138 31 107)(18 139 32 108)(19 140 33 97)(20 141 34 98)(21 142 35 99)(22 143 36 100)(23 144 25 101)(24 133 26 102)(37 163 145 125)(38 164 146 126)(39 165 147 127)(40 166 148 128)(41 167 149 129)(42 168 150 130)(43 157 151 131)(44 158 152 132)(45 159 153 121)(46 160 154 122)(47 161 155 123)(48 162 156 124)(61 120 90 191)(62 109 91 192)(63 110 92 181)(64 111 93 182)(65 112 94 183)(66 113 95 184)(67 114 96 185)(68 115 85 186)(69 116 86 187)(70 117 87 188)(71 118 88 189)(72 119 89 190)
(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 87 7 93)(2 86 8 92)(3 85 9 91)(4 96 10 90)(5 95 11 89)(6 94 12 88)(13 46 19 40)(14 45 20 39)(15 44 21 38)(16 43 22 37)(17 42 23 48)(18 41 24 47)(25 156 31 150)(26 155 32 149)(27 154 33 148)(28 153 34 147)(29 152 35 146)(30 151 36 145)(49 185 55 191)(50 184 56 190)(51 183 57 189)(52 182 58 188)(53 181 59 187)(54 192 60 186)(61 171 67 177)(62 170 68 176)(63 169 69 175)(64 180 70 174)(65 179 71 173)(66 178 72 172)(73 110 79 116)(74 109 80 115)(75 120 81 114)(76 119 82 113)(77 118 83 112)(78 117 84 111)(97 128 103 122)(98 127 104 121)(99 126 105 132)(100 125 106 131)(101 124 107 130)(102 123 108 129)(133 161 139 167)(134 160 140 166)(135 159 141 165)(136 158 142 164)(137 157 143 163)(138 168 144 162)
G:=sub<Sym(192)| (1,43,180,151)(2,44,169,152)(3,45,170,153)(4,46,171,154)(5,47,172,155)(6,48,173,156)(7,37,174,145)(8,38,175,146)(9,39,176,147)(10,40,177,148)(11,41,178,149)(12,42,179,150)(13,61,27,90)(14,62,28,91)(15,63,29,92)(16,64,30,93)(17,65,31,94)(18,66,32,95)(19,67,33,96)(20,68,34,85)(21,69,35,86)(22,70,36,87)(23,71,25,88)(24,72,26,89)(49,122,81,160)(50,123,82,161)(51,124,83,162)(52,125,84,163)(53,126,73,164)(54,127,74,165)(55,128,75,166)(56,129,76,167)(57,130,77,168)(58,131,78,157)(59,132,79,158)(60,121,80,159)(97,114,140,185)(98,115,141,186)(99,116,142,187)(100,117,143,188)(101,118,144,189)(102,119,133,190)(103,120,134,191)(104,109,135,192)(105,110,136,181)(106,111,137,182)(107,112,138,183)(108,113,139,184), (1,58,180,78)(2,59,169,79)(3,60,170,80)(4,49,171,81)(5,50,172,82)(6,51,173,83)(7,52,174,84)(8,53,175,73)(9,54,176,74)(10,55,177,75)(11,56,178,76)(12,57,179,77)(13,134,27,103)(14,135,28,104)(15,136,29,105)(16,137,30,106)(17,138,31,107)(18,139,32,108)(19,140,33,97)(20,141,34,98)(21,142,35,99)(22,143,36,100)(23,144,25,101)(24,133,26,102)(37,163,145,125)(38,164,146,126)(39,165,147,127)(40,166,148,128)(41,167,149,129)(42,168,150,130)(43,157,151,131)(44,158,152,132)(45,159,153,121)(46,160,154,122)(47,161,155,123)(48,162,156,124)(61,120,90,191)(62,109,91,192)(63,110,92,181)(64,111,93,182)(65,112,94,183)(66,113,95,184)(67,114,96,185)(68,115,85,186)(69,116,86,187)(70,117,87,188)(71,118,88,189)(72,119,89,190), (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), (1,87,7,93)(2,86,8,92)(3,85,9,91)(4,96,10,90)(5,95,11,89)(6,94,12,88)(13,46,19,40)(14,45,20,39)(15,44,21,38)(16,43,22,37)(17,42,23,48)(18,41,24,47)(25,156,31,150)(26,155,32,149)(27,154,33,148)(28,153,34,147)(29,152,35,146)(30,151,36,145)(49,185,55,191)(50,184,56,190)(51,183,57,189)(52,182,58,188)(53,181,59,187)(54,192,60,186)(61,171,67,177)(62,170,68,176)(63,169,69,175)(64,180,70,174)(65,179,71,173)(66,178,72,172)(73,110,79,116)(74,109,80,115)(75,120,81,114)(76,119,82,113)(77,118,83,112)(78,117,84,111)(97,128,103,122)(98,127,104,121)(99,126,105,132)(100,125,106,131)(101,124,107,130)(102,123,108,129)(133,161,139,167)(134,160,140,166)(135,159,141,165)(136,158,142,164)(137,157,143,163)(138,168,144,162)>;
G:=Group( (1,43,180,151)(2,44,169,152)(3,45,170,153)(4,46,171,154)(5,47,172,155)(6,48,173,156)(7,37,174,145)(8,38,175,146)(9,39,176,147)(10,40,177,148)(11,41,178,149)(12,42,179,150)(13,61,27,90)(14,62,28,91)(15,63,29,92)(16,64,30,93)(17,65,31,94)(18,66,32,95)(19,67,33,96)(20,68,34,85)(21,69,35,86)(22,70,36,87)(23,71,25,88)(24,72,26,89)(49,122,81,160)(50,123,82,161)(51,124,83,162)(52,125,84,163)(53,126,73,164)(54,127,74,165)(55,128,75,166)(56,129,76,167)(57,130,77,168)(58,131,78,157)(59,132,79,158)(60,121,80,159)(97,114,140,185)(98,115,141,186)(99,116,142,187)(100,117,143,188)(101,118,144,189)(102,119,133,190)(103,120,134,191)(104,109,135,192)(105,110,136,181)(106,111,137,182)(107,112,138,183)(108,113,139,184), (1,58,180,78)(2,59,169,79)(3,60,170,80)(4,49,171,81)(5,50,172,82)(6,51,173,83)(7,52,174,84)(8,53,175,73)(9,54,176,74)(10,55,177,75)(11,56,178,76)(12,57,179,77)(13,134,27,103)(14,135,28,104)(15,136,29,105)(16,137,30,106)(17,138,31,107)(18,139,32,108)(19,140,33,97)(20,141,34,98)(21,142,35,99)(22,143,36,100)(23,144,25,101)(24,133,26,102)(37,163,145,125)(38,164,146,126)(39,165,147,127)(40,166,148,128)(41,167,149,129)(42,168,150,130)(43,157,151,131)(44,158,152,132)(45,159,153,121)(46,160,154,122)(47,161,155,123)(48,162,156,124)(61,120,90,191)(62,109,91,192)(63,110,92,181)(64,111,93,182)(65,112,94,183)(66,113,95,184)(67,114,96,185)(68,115,85,186)(69,116,86,187)(70,117,87,188)(71,118,88,189)(72,119,89,190), (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), (1,87,7,93)(2,86,8,92)(3,85,9,91)(4,96,10,90)(5,95,11,89)(6,94,12,88)(13,46,19,40)(14,45,20,39)(15,44,21,38)(16,43,22,37)(17,42,23,48)(18,41,24,47)(25,156,31,150)(26,155,32,149)(27,154,33,148)(28,153,34,147)(29,152,35,146)(30,151,36,145)(49,185,55,191)(50,184,56,190)(51,183,57,189)(52,182,58,188)(53,181,59,187)(54,192,60,186)(61,171,67,177)(62,170,68,176)(63,169,69,175)(64,180,70,174)(65,179,71,173)(66,178,72,172)(73,110,79,116)(74,109,80,115)(75,120,81,114)(76,119,82,113)(77,118,83,112)(78,117,84,111)(97,128,103,122)(98,127,104,121)(99,126,105,132)(100,125,106,131)(101,124,107,130)(102,123,108,129)(133,161,139,167)(134,160,140,166)(135,159,141,165)(136,158,142,164)(137,157,143,163)(138,168,144,162) );
G=PermutationGroup([(1,43,180,151),(2,44,169,152),(3,45,170,153),(4,46,171,154),(5,47,172,155),(6,48,173,156),(7,37,174,145),(8,38,175,146),(9,39,176,147),(10,40,177,148),(11,41,178,149),(12,42,179,150),(13,61,27,90),(14,62,28,91),(15,63,29,92),(16,64,30,93),(17,65,31,94),(18,66,32,95),(19,67,33,96),(20,68,34,85),(21,69,35,86),(22,70,36,87),(23,71,25,88),(24,72,26,89),(49,122,81,160),(50,123,82,161),(51,124,83,162),(52,125,84,163),(53,126,73,164),(54,127,74,165),(55,128,75,166),(56,129,76,167),(57,130,77,168),(58,131,78,157),(59,132,79,158),(60,121,80,159),(97,114,140,185),(98,115,141,186),(99,116,142,187),(100,117,143,188),(101,118,144,189),(102,119,133,190),(103,120,134,191),(104,109,135,192),(105,110,136,181),(106,111,137,182),(107,112,138,183),(108,113,139,184)], [(1,58,180,78),(2,59,169,79),(3,60,170,80),(4,49,171,81),(5,50,172,82),(6,51,173,83),(7,52,174,84),(8,53,175,73),(9,54,176,74),(10,55,177,75),(11,56,178,76),(12,57,179,77),(13,134,27,103),(14,135,28,104),(15,136,29,105),(16,137,30,106),(17,138,31,107),(18,139,32,108),(19,140,33,97),(20,141,34,98),(21,142,35,99),(22,143,36,100),(23,144,25,101),(24,133,26,102),(37,163,145,125),(38,164,146,126),(39,165,147,127),(40,166,148,128),(41,167,149,129),(42,168,150,130),(43,157,151,131),(44,158,152,132),(45,159,153,121),(46,160,154,122),(47,161,155,123),(48,162,156,124),(61,120,90,191),(62,109,91,192),(63,110,92,181),(64,111,93,182),(65,112,94,183),(66,113,95,184),(67,114,96,185),(68,115,85,186),(69,116,86,187),(70,117,87,188),(71,118,88,189),(72,119,89,190)], [(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)], [(1,87,7,93),(2,86,8,92),(3,85,9,91),(4,96,10,90),(5,95,11,89),(6,94,12,88),(13,46,19,40),(14,45,20,39),(15,44,21,38),(16,43,22,37),(17,42,23,48),(18,41,24,47),(25,156,31,150),(26,155,32,149),(27,154,33,148),(28,153,34,147),(29,152,35,146),(30,151,36,145),(49,185,55,191),(50,184,56,190),(51,183,57,189),(52,182,58,188),(53,181,59,187),(54,192,60,186),(61,171,67,177),(62,170,68,176),(63,169,69,175),(64,180,70,174),(65,179,71,173),(66,178,72,172),(73,110,79,116),(74,109,80,115),(75,120,81,114),(76,119,82,113),(77,118,83,112),(78,117,84,111),(97,128,103,122),(98,127,104,121),(99,126,105,132),(100,125,106,131),(101,124,107,130),(102,123,108,129),(133,161,139,167),(134,160,140,166),(135,159,141,165),(136,158,142,164),(137,157,143,163),(138,168,144,162)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 8 | 11 |
| 0 | 0 | 0 | 5 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 1 | 8 |
| 6 | 3 | 0 | 0 |
| 10 | 3 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 2 | 11 | 0 | 0 |
| 9 | 11 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,8,0,0,0,11,5],[1,0,0,0,0,1,0,0,0,0,5,1,0,0,0,8],[6,10,0,0,3,3,0,0,0,0,12,0,0,0,0,12],[2,9,0,0,11,11,0,0,0,0,12,0,0,0,0,12] >;
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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | - | + | + | + | - | + | - | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | Q8 | D6 | D6 | D6 | Dic6 | 2+ (1+4) | S3×Q8 | D4○D12 |
| kernel | Q8×Dic6 | C4×Dic6 | C12⋊2Q8 | C12⋊Q8 | Q8×Dic3 | Q8×C12 | C4×Q8 | Dic6 | C3×Q8 | C42 | C4⋊C4 | C2×Q8 | Q8 | C6 | C4 | C2 |
| # reps | 1 | 3 | 3 | 6 | 2 | 1 | 1 | 4 | 4 | 3 | 3 | 1 | 8 | 1 | 2 | 2 |
In GAP, Magma, Sage, TeX
Q_8\times Dic_6
% in TeX
G:=Group("Q8xDic6"); // GroupNames label
G:=SmallGroup(192,1125);
// by ID
G=gap.SmallGroup(192,1125);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,232,387,184,675,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^12=1,b^2=a^2,d^2=c^6,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀