direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C2×C12, (C2×C42).20C6, C42.86(C2×C6), C6.57(C23×C4), C2.5(C23×C12), C6.56(C22×Q8), C22.17(C6×Q8), C4.17(C22×C12), (C2×C6).336C24, C22.9(C23×C6), (C22×Q8).17C6, (C4×C12).370C22, (C2×C12).708C23, C12.162(C22×C4), C23.72(C22×C6), (C6×Q8).282C22, (C22×C6).468C23, C22.26(C22×C12), (C22×C12).594C22, C2.2(Q8×C2×C6), (C2×C4×C12).40C2, C2.3(C6×C4○D4), (C2×C4⋊C4).22C6, (C6×C4⋊C4).51C2, (Q8×C2×C6).20C2, C4⋊C4.80(C2×C6), (C2×C4).52(C2×C12), C6.222(C2×C4○D4), (C2×Q8).82(C2×C6), (C2×C6).115(C2×Q8), (C2×C12).273(C2×C4), (C2×C4).55(C22×C6), C22.28(C3×C4○D4), (C2×C6).228(C4○D4), (C3×C4⋊C4).405C22, (C2×C6).246(C22×C4), (C22×C4).105(C2×C6), SmallGroup(192,1405)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 322 in 298 conjugacy classes, 274 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×16], C4 [×6], C22, C22 [×6], C6 [×3], C6 [×4], C2×C4 [×30], C2×C4 [×6], Q8 [×16], C23, C12 [×16], C12 [×6], C2×C6, C2×C6 [×6], C42 [×12], C4⋊C4 [×12], C22×C4, C22×C4 [×6], C2×Q8 [×12], C2×C12 [×30], C2×C12 [×6], C3×Q8 [×16], C22×C6, C2×C42 [×3], C2×C4⋊C4 [×3], C4×Q8 [×8], C22×Q8, C4×C12 [×12], C3×C4⋊C4 [×12], C22×C12, C22×C12 [×6], C6×Q8 [×12], C2×C4×Q8, C2×C4×C12 [×3], C6×C4⋊C4 [×3], Q8×C12 [×8], Q8×C2×C6, Q8×C2×C12
Quotients:
C1, C2 [×15], C3, C4 [×8], C22 [×35], C6 [×15], C2×C4 [×28], Q8 [×4], C23 [×15], C12 [×8], C2×C6 [×35], C22×C4 [×14], C2×Q8 [×6], C4○D4 [×2], C24, C2×C12 [×28], C3×Q8 [×4], C22×C6 [×15], C4×Q8 [×4], C23×C4, C22×Q8, C2×C4○D4, C22×C12 [×14], C6×Q8 [×6], C3×C4○D4 [×2], C23×C6, C2×C4×Q8, Q8×C12 [×4], C23×C12, Q8×C2×C6, C6×C4○D4, Q8×C2×C12
Generators and relations
G = < a,b,c,d | a2=b12=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
►Regular action on 192 points
Generators in S192
(1 14)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 13)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(31 49)(32 50)(33 51)(34 52)(35 53)(36 54)(37 103)(38 104)(39 105)(40 106)(41 107)(42 108)(43 97)(44 98)(45 99)(46 100)(47 101)(48 102)(61 127)(62 128)(63 129)(64 130)(65 131)(66 132)(67 121)(68 122)(69 123)(70 124)(71 125)(72 126)(73 180)(74 169)(75 170)(76 171)(77 172)(78 173)(79 174)(80 175)(81 176)(82 177)(83 178)(84 179)(85 183)(86 184)(87 185)(88 186)(89 187)(90 188)(91 189)(92 190)(93 191)(94 192)(95 181)(96 182)(109 141)(110 142)(111 143)(112 144)(113 133)(114 134)(115 135)(116 136)(117 137)(118 138)(119 139)(120 140)(145 166)(146 167)(147 168)(148 157)(149 158)(150 159)(151 160)(152 161)(153 162)(154 163)(155 164)(156 165)
(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 73 110 101)(2 74 111 102)(3 75 112 103)(4 76 113 104)(5 77 114 105)(6 78 115 106)(7 79 116 107)(8 80 117 108)(9 81 118 97)(10 82 119 98)(11 83 120 99)(12 84 109 100)(13 179 141 46)(14 180 142 47)(15 169 143 48)(16 170 144 37)(17 171 133 38)(18 172 134 39)(19 173 135 40)(20 174 136 41)(21 175 137 42)(22 176 138 43)(23 177 139 44)(24 178 140 45)(25 93 65 160)(26 94 66 161)(27 95 67 162)(28 96 68 163)(29 85 69 164)(30 86 70 165)(31 87 71 166)(32 88 72 167)(33 89 61 168)(34 90 62 157)(35 91 63 158)(36 92 64 159)(49 185 125 145)(50 186 126 146)(51 187 127 147)(52 188 128 148)(53 189 129 149)(54 190 130 150)(55 191 131 151)(56 192 132 152)(57 181 121 153)(58 182 122 154)(59 183 123 155)(60 184 124 156)
(1 153 110 181)(2 154 111 182)(3 155 112 183)(4 156 113 184)(5 145 114 185)(6 146 115 186)(7 147 116 187)(8 148 117 188)(9 149 118 189)(10 150 119 190)(11 151 120 191)(12 152 109 192)(13 161 141 94)(14 162 142 95)(15 163 143 96)(16 164 144 85)(17 165 133 86)(18 166 134 87)(19 167 135 88)(20 168 136 89)(21 157 137 90)(22 158 138 91)(23 159 139 92)(24 160 140 93)(25 178 65 45)(26 179 66 46)(27 180 67 47)(28 169 68 48)(29 170 69 37)(30 171 70 38)(31 172 71 39)(32 173 72 40)(33 174 61 41)(34 175 62 42)(35 176 63 43)(36 177 64 44)(49 77 125 105)(50 78 126 106)(51 79 127 107)(52 80 128 108)(53 81 129 97)(54 82 130 98)(55 83 131 99)(56 84 132 100)(57 73 121 101)(58 74 122 102)(59 75 123 103)(60 76 124 104)
G:=sub<Sym(192)| (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,13)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,49)(32,50)(33,51)(34,52)(35,53)(36,54)(37,103)(38,104)(39,105)(40,106)(41,107)(42,108)(43,97)(44,98)(45,99)(46,100)(47,101)(48,102)(61,127)(62,128)(63,129)(64,130)(65,131)(66,132)(67,121)(68,122)(69,123)(70,124)(71,125)(72,126)(73,180)(74,169)(75,170)(76,171)(77,172)(78,173)(79,174)(80,175)(81,176)(82,177)(83,178)(84,179)(85,183)(86,184)(87,185)(88,186)(89,187)(90,188)(91,189)(92,190)(93,191)(94,192)(95,181)(96,182)(109,141)(110,142)(111,143)(112,144)(113,133)(114,134)(115,135)(116,136)(117,137)(118,138)(119,139)(120,140)(145,166)(146,167)(147,168)(148,157)(149,158)(150,159)(151,160)(152,161)(153,162)(154,163)(155,164)(156,165), (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), (1,73,110,101)(2,74,111,102)(3,75,112,103)(4,76,113,104)(5,77,114,105)(6,78,115,106)(7,79,116,107)(8,80,117,108)(9,81,118,97)(10,82,119,98)(11,83,120,99)(12,84,109,100)(13,179,141,46)(14,180,142,47)(15,169,143,48)(16,170,144,37)(17,171,133,38)(18,172,134,39)(19,173,135,40)(20,174,136,41)(21,175,137,42)(22,176,138,43)(23,177,139,44)(24,178,140,45)(25,93,65,160)(26,94,66,161)(27,95,67,162)(28,96,68,163)(29,85,69,164)(30,86,70,165)(31,87,71,166)(32,88,72,167)(33,89,61,168)(34,90,62,157)(35,91,63,158)(36,92,64,159)(49,185,125,145)(50,186,126,146)(51,187,127,147)(52,188,128,148)(53,189,129,149)(54,190,130,150)(55,191,131,151)(56,192,132,152)(57,181,121,153)(58,182,122,154)(59,183,123,155)(60,184,124,156), (1,153,110,181)(2,154,111,182)(3,155,112,183)(4,156,113,184)(5,145,114,185)(6,146,115,186)(7,147,116,187)(8,148,117,188)(9,149,118,189)(10,150,119,190)(11,151,120,191)(12,152,109,192)(13,161,141,94)(14,162,142,95)(15,163,143,96)(16,164,144,85)(17,165,133,86)(18,166,134,87)(19,167,135,88)(20,168,136,89)(21,157,137,90)(22,158,138,91)(23,159,139,92)(24,160,140,93)(25,178,65,45)(26,179,66,46)(27,180,67,47)(28,169,68,48)(29,170,69,37)(30,171,70,38)(31,172,71,39)(32,173,72,40)(33,174,61,41)(34,175,62,42)(35,176,63,43)(36,177,64,44)(49,77,125,105)(50,78,126,106)(51,79,127,107)(52,80,128,108)(53,81,129,97)(54,82,130,98)(55,83,131,99)(56,84,132,100)(57,73,121,101)(58,74,122,102)(59,75,123,103)(60,76,124,104)>;
G:=Group( (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,13)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,49)(32,50)(33,51)(34,52)(35,53)(36,54)(37,103)(38,104)(39,105)(40,106)(41,107)(42,108)(43,97)(44,98)(45,99)(46,100)(47,101)(48,102)(61,127)(62,128)(63,129)(64,130)(65,131)(66,132)(67,121)(68,122)(69,123)(70,124)(71,125)(72,126)(73,180)(74,169)(75,170)(76,171)(77,172)(78,173)(79,174)(80,175)(81,176)(82,177)(83,178)(84,179)(85,183)(86,184)(87,185)(88,186)(89,187)(90,188)(91,189)(92,190)(93,191)(94,192)(95,181)(96,182)(109,141)(110,142)(111,143)(112,144)(113,133)(114,134)(115,135)(116,136)(117,137)(118,138)(119,139)(120,140)(145,166)(146,167)(147,168)(148,157)(149,158)(150,159)(151,160)(152,161)(153,162)(154,163)(155,164)(156,165), (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), (1,73,110,101)(2,74,111,102)(3,75,112,103)(4,76,113,104)(5,77,114,105)(6,78,115,106)(7,79,116,107)(8,80,117,108)(9,81,118,97)(10,82,119,98)(11,83,120,99)(12,84,109,100)(13,179,141,46)(14,180,142,47)(15,169,143,48)(16,170,144,37)(17,171,133,38)(18,172,134,39)(19,173,135,40)(20,174,136,41)(21,175,137,42)(22,176,138,43)(23,177,139,44)(24,178,140,45)(25,93,65,160)(26,94,66,161)(27,95,67,162)(28,96,68,163)(29,85,69,164)(30,86,70,165)(31,87,71,166)(32,88,72,167)(33,89,61,168)(34,90,62,157)(35,91,63,158)(36,92,64,159)(49,185,125,145)(50,186,126,146)(51,187,127,147)(52,188,128,148)(53,189,129,149)(54,190,130,150)(55,191,131,151)(56,192,132,152)(57,181,121,153)(58,182,122,154)(59,183,123,155)(60,184,124,156), (1,153,110,181)(2,154,111,182)(3,155,112,183)(4,156,113,184)(5,145,114,185)(6,146,115,186)(7,147,116,187)(8,148,117,188)(9,149,118,189)(10,150,119,190)(11,151,120,191)(12,152,109,192)(13,161,141,94)(14,162,142,95)(15,163,143,96)(16,164,144,85)(17,165,133,86)(18,166,134,87)(19,167,135,88)(20,168,136,89)(21,157,137,90)(22,158,138,91)(23,159,139,92)(24,160,140,93)(25,178,65,45)(26,179,66,46)(27,180,67,47)(28,169,68,48)(29,170,69,37)(30,171,70,38)(31,172,71,39)(32,173,72,40)(33,174,61,41)(34,175,62,42)(35,176,63,43)(36,177,64,44)(49,77,125,105)(50,78,126,106)(51,79,127,107)(52,80,128,108)(53,81,129,97)(54,82,130,98)(55,83,131,99)(56,84,132,100)(57,73,121,101)(58,74,122,102)(59,75,123,103)(60,76,124,104) );
G=PermutationGroup([(1,14),(2,15),(3,16),(4,17),(5,18),(6,19),(7,20),(8,21),(9,22),(10,23),(11,24),(12,13),(25,55),(26,56),(27,57),(28,58),(29,59),(30,60),(31,49),(32,50),(33,51),(34,52),(35,53),(36,54),(37,103),(38,104),(39,105),(40,106),(41,107),(42,108),(43,97),(44,98),(45,99),(46,100),(47,101),(48,102),(61,127),(62,128),(63,129),(64,130),(65,131),(66,132),(67,121),(68,122),(69,123),(70,124),(71,125),(72,126),(73,180),(74,169),(75,170),(76,171),(77,172),(78,173),(79,174),(80,175),(81,176),(82,177),(83,178),(84,179),(85,183),(86,184),(87,185),(88,186),(89,187),(90,188),(91,189),(92,190),(93,191),(94,192),(95,181),(96,182),(109,141),(110,142),(111,143),(112,144),(113,133),(114,134),(115,135),(116,136),(117,137),(118,138),(119,139),(120,140),(145,166),(146,167),(147,168),(148,157),(149,158),(150,159),(151,160),(152,161),(153,162),(154,163),(155,164),(156,165)], [(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)], [(1,73,110,101),(2,74,111,102),(3,75,112,103),(4,76,113,104),(5,77,114,105),(6,78,115,106),(7,79,116,107),(8,80,117,108),(9,81,118,97),(10,82,119,98),(11,83,120,99),(12,84,109,100),(13,179,141,46),(14,180,142,47),(15,169,143,48),(16,170,144,37),(17,171,133,38),(18,172,134,39),(19,173,135,40),(20,174,136,41),(21,175,137,42),(22,176,138,43),(23,177,139,44),(24,178,140,45),(25,93,65,160),(26,94,66,161),(27,95,67,162),(28,96,68,163),(29,85,69,164),(30,86,70,165),(31,87,71,166),(32,88,72,167),(33,89,61,168),(34,90,62,157),(35,91,63,158),(36,92,64,159),(49,185,125,145),(50,186,126,146),(51,187,127,147),(52,188,128,148),(53,189,129,149),(54,190,130,150),(55,191,131,151),(56,192,132,152),(57,181,121,153),(58,182,122,154),(59,183,123,155),(60,184,124,156)], [(1,153,110,181),(2,154,111,182),(3,155,112,183),(4,156,113,184),(5,145,114,185),(6,146,115,186),(7,147,116,187),(8,148,117,188),(9,149,118,189),(10,150,119,190),(11,151,120,191),(12,152,109,192),(13,161,141,94),(14,162,142,95),(15,163,143,96),(16,164,144,85),(17,165,133,86),(18,166,134,87),(19,167,135,88),(20,168,136,89),(21,157,137,90),(22,158,138,91),(23,159,139,92),(24,160,140,93),(25,178,65,45),(26,179,66,46),(27,180,67,47),(28,169,68,48),(29,170,69,37),(30,171,70,38),(31,172,71,39),(32,173,72,40),(33,174,61,41),(34,175,62,42),(35,176,63,43),(36,177,64,44),(49,77,125,105),(50,78,126,106),(51,79,127,107),(52,80,128,108),(53,81,129,97),(54,82,130,98),(55,83,131,99),(56,84,132,100),(57,73,121,101),(58,74,122,102),(59,75,123,103),(60,76,124,104)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 12 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 10 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 3 | 9 |
| 0 | 0 | 9 | 10 |
G:=sub<GL(4,GF(13))| [12,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[10,0,0,0,0,8,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,12,0,0,0,0,0,12,0,0,1,0],[12,0,0,0,0,12,0,0,0,0,3,9,0,0,9,10] >;
120 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | 3B | 4A | ··· | 4H | 4I | ··· | 4AF | 6A | ··· | 6N | 12A | ··· | 12P | 12Q | ··· | 12BL |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C12 | Q8 | C4○D4 | C3×Q8 | C3×C4○D4 |
| kernel | Q8×C2×C12 | C2×C4×C12 | C6×C4⋊C4 | Q8×C12 | Q8×C2×C6 | C2×C4×Q8 | C6×Q8 | C2×C42 | C2×C4⋊C4 | C4×Q8 | C22×Q8 | C2×Q8 | C2×C12 | C2×C6 | C2×C4 | C22 |
| # reps | 1 | 3 | 3 | 8 | 1 | 2 | 16 | 6 | 6 | 16 | 2 | 32 | 4 | 4 | 8 | 8 |
In GAP, Magma, Sage, TeX
Q_8\times C_2\times C_{12} % in TeX
G:=Group("Q8xC2xC12"); // GroupNames label
G:=SmallGroup(192,1405);
// by ID
G=gap.SmallGroup(192,1405);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,672,701,344,772]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=c^4=1,d^2=c^2,a*b=b*a,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
○
≀