direct product, abelian, monomial, 2-elementary
Aliases: C22×C4×C12, SmallGroup(192,1400)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C22×C4×C12 |
| C1 — C22×C4×C12 |
| C1 — C22×C4×C12 |
Subgroups: 498, all normal (8 characteristic)
C1, C2 [×15], C3, C4 [×24], C22, C22 [×34], C6 [×15], C2×C4 [×84], C23 [×15], C12 [×24], C2×C6, C2×C6 [×34], C42 [×16], C22×C4 [×42], C24, C2×C12 [×84], C22×C6 [×15], C2×C42 [×12], C23×C4 [×3], C4×C12 [×16], C22×C12 [×42], C23×C6, C22×C42, C2×C4×C12 [×12], C23×C12 [×3], C22×C4×C12
Quotients:
C1, C2 [×15], C3, C4 [×24], C22 [×35], C6 [×15], C2×C4 [×84], C23 [×15], C12 [×24], C2×C6 [×35], C42 [×16], C22×C4 [×42], C24, C2×C12 [×84], C22×C6 [×15], C2×C42 [×12], C23×C4 [×3], C4×C12 [×16], C22×C12 [×42], C23×C6, C22×C42, C2×C4×C12 [×12], C23×C12 [×3], C22×C4×C12
Generators and relations
G = < a,b,c,d | a2=b2=c4=d12=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >
►Regular action on 192 points
Generators in S192
(1 148)(2 149)(3 150)(4 151)(5 152)(6 153)(7 154)(8 155)(9 156)(10 145)(11 146)(12 147)(13 92)(14 93)(15 94)(16 95)(17 96)(18 85)(19 86)(20 87)(21 88)(22 89)(23 90)(24 91)(25 181)(26 182)(27 183)(28 184)(29 185)(30 186)(31 187)(32 188)(33 189)(34 190)(35 191)(36 192)(37 105)(38 106)(39 107)(40 108)(41 97)(42 98)(43 99)(44 100)(45 101)(46 102)(47 103)(48 104)(49 69)(50 70)(51 71)(52 72)(53 61)(54 62)(55 63)(56 64)(57 65)(58 66)(59 67)(60 68)(73 165)(74 166)(75 167)(76 168)(77 157)(78 158)(79 159)(80 160)(81 161)(82 162)(83 163)(84 164)(109 129)(110 130)(111 131)(112 132)(113 121)(114 122)(115 123)(116 124)(117 125)(118 126)(119 127)(120 128)(133 171)(134 172)(135 173)(136 174)(137 175)(138 176)(139 177)(140 178)(141 179)(142 180)(143 169)(144 170)
(1 23)(2 24)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(11 21)(12 22)(25 164)(26 165)(27 166)(28 167)(29 168)(30 157)(31 158)(32 159)(33 160)(34 161)(35 162)(36 163)(37 128)(38 129)(39 130)(40 131)(41 132)(42 121)(43 122)(44 123)(45 124)(46 125)(47 126)(48 127)(49 138)(50 139)(51 140)(52 141)(53 142)(54 143)(55 144)(56 133)(57 134)(58 135)(59 136)(60 137)(61 180)(62 169)(63 170)(64 171)(65 172)(66 173)(67 174)(68 175)(69 176)(70 177)(71 178)(72 179)(73 182)(74 183)(75 184)(76 185)(77 186)(78 187)(79 188)(80 189)(81 190)(82 191)(83 192)(84 181)(85 155)(86 156)(87 145)(88 146)(89 147)(90 148)(91 149)(92 150)(93 151)(94 152)(95 153)(96 154)(97 112)(98 113)(99 114)(100 115)(101 116)(102 117)(103 118)(104 119)(105 120)(106 109)(107 110)(108 111)
(1 116 74 71)(2 117 75 72)(3 118 76 61)(4 119 77 62)(5 120 78 63)(6 109 79 64)(7 110 80 65)(8 111 81 66)(9 112 82 67)(10 113 83 68)(11 114 84 69)(12 115 73 70)(13 103 185 180)(14 104 186 169)(15 105 187 170)(16 106 188 171)(17 107 189 172)(18 108 190 173)(19 97 191 174)(20 98 192 175)(21 99 181 176)(22 100 182 177)(23 101 183 178)(24 102 184 179)(25 138 88 43)(26 139 89 44)(27 140 90 45)(28 141 91 46)(29 142 92 47)(30 143 93 48)(31 144 94 37)(32 133 95 38)(33 134 96 39)(34 135 85 40)(35 136 86 41)(36 137 87 42)(49 146 122 164)(50 147 123 165)(51 148 124 166)(52 149 125 167)(53 150 126 168)(54 151 127 157)(55 152 128 158)(56 153 129 159)(57 154 130 160)(58 155 131 161)(59 156 132 162)(60 145 121 163)
(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)
G:=sub<Sym(192)| (1,148)(2,149)(3,150)(4,151)(5,152)(6,153)(7,154)(8,155)(9,156)(10,145)(11,146)(12,147)(13,92)(14,93)(15,94)(16,95)(17,96)(18,85)(19,86)(20,87)(21,88)(22,89)(23,90)(24,91)(25,181)(26,182)(27,183)(28,184)(29,185)(30,186)(31,187)(32,188)(33,189)(34,190)(35,191)(36,192)(37,105)(38,106)(39,107)(40,108)(41,97)(42,98)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,69)(50,70)(51,71)(52,72)(53,61)(54,62)(55,63)(56,64)(57,65)(58,66)(59,67)(60,68)(73,165)(74,166)(75,167)(76,168)(77,157)(78,158)(79,159)(80,160)(81,161)(82,162)(83,163)(84,164)(109,129)(110,130)(111,131)(112,132)(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,127)(120,128)(133,171)(134,172)(135,173)(136,174)(137,175)(138,176)(139,177)(140,178)(141,179)(142,180)(143,169)(144,170), (1,23)(2,24)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(11,21)(12,22)(25,164)(26,165)(27,166)(28,167)(29,168)(30,157)(31,158)(32,159)(33,160)(34,161)(35,162)(36,163)(37,128)(38,129)(39,130)(40,131)(41,132)(42,121)(43,122)(44,123)(45,124)(46,125)(47,126)(48,127)(49,138)(50,139)(51,140)(52,141)(53,142)(54,143)(55,144)(56,133)(57,134)(58,135)(59,136)(60,137)(61,180)(62,169)(63,170)(64,171)(65,172)(66,173)(67,174)(68,175)(69,176)(70,177)(71,178)(72,179)(73,182)(74,183)(75,184)(76,185)(77,186)(78,187)(79,188)(80,189)(81,190)(82,191)(83,192)(84,181)(85,155)(86,156)(87,145)(88,146)(89,147)(90,148)(91,149)(92,150)(93,151)(94,152)(95,153)(96,154)(97,112)(98,113)(99,114)(100,115)(101,116)(102,117)(103,118)(104,119)(105,120)(106,109)(107,110)(108,111), (1,116,74,71)(2,117,75,72)(3,118,76,61)(4,119,77,62)(5,120,78,63)(6,109,79,64)(7,110,80,65)(8,111,81,66)(9,112,82,67)(10,113,83,68)(11,114,84,69)(12,115,73,70)(13,103,185,180)(14,104,186,169)(15,105,187,170)(16,106,188,171)(17,107,189,172)(18,108,190,173)(19,97,191,174)(20,98,192,175)(21,99,181,176)(22,100,182,177)(23,101,183,178)(24,102,184,179)(25,138,88,43)(26,139,89,44)(27,140,90,45)(28,141,91,46)(29,142,92,47)(30,143,93,48)(31,144,94,37)(32,133,95,38)(33,134,96,39)(34,135,85,40)(35,136,86,41)(36,137,87,42)(49,146,122,164)(50,147,123,165)(51,148,124,166)(52,149,125,167)(53,150,126,168)(54,151,127,157)(55,152,128,158)(56,153,129,159)(57,154,130,160)(58,155,131,161)(59,156,132,162)(60,145,121,163), (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)>;
G:=Group( (1,148)(2,149)(3,150)(4,151)(5,152)(6,153)(7,154)(8,155)(9,156)(10,145)(11,146)(12,147)(13,92)(14,93)(15,94)(16,95)(17,96)(18,85)(19,86)(20,87)(21,88)(22,89)(23,90)(24,91)(25,181)(26,182)(27,183)(28,184)(29,185)(30,186)(31,187)(32,188)(33,189)(34,190)(35,191)(36,192)(37,105)(38,106)(39,107)(40,108)(41,97)(42,98)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,69)(50,70)(51,71)(52,72)(53,61)(54,62)(55,63)(56,64)(57,65)(58,66)(59,67)(60,68)(73,165)(74,166)(75,167)(76,168)(77,157)(78,158)(79,159)(80,160)(81,161)(82,162)(83,163)(84,164)(109,129)(110,130)(111,131)(112,132)(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,127)(120,128)(133,171)(134,172)(135,173)(136,174)(137,175)(138,176)(139,177)(140,178)(141,179)(142,180)(143,169)(144,170), (1,23)(2,24)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(11,21)(12,22)(25,164)(26,165)(27,166)(28,167)(29,168)(30,157)(31,158)(32,159)(33,160)(34,161)(35,162)(36,163)(37,128)(38,129)(39,130)(40,131)(41,132)(42,121)(43,122)(44,123)(45,124)(46,125)(47,126)(48,127)(49,138)(50,139)(51,140)(52,141)(53,142)(54,143)(55,144)(56,133)(57,134)(58,135)(59,136)(60,137)(61,180)(62,169)(63,170)(64,171)(65,172)(66,173)(67,174)(68,175)(69,176)(70,177)(71,178)(72,179)(73,182)(74,183)(75,184)(76,185)(77,186)(78,187)(79,188)(80,189)(81,190)(82,191)(83,192)(84,181)(85,155)(86,156)(87,145)(88,146)(89,147)(90,148)(91,149)(92,150)(93,151)(94,152)(95,153)(96,154)(97,112)(98,113)(99,114)(100,115)(101,116)(102,117)(103,118)(104,119)(105,120)(106,109)(107,110)(108,111), (1,116,74,71)(2,117,75,72)(3,118,76,61)(4,119,77,62)(5,120,78,63)(6,109,79,64)(7,110,80,65)(8,111,81,66)(9,112,82,67)(10,113,83,68)(11,114,84,69)(12,115,73,70)(13,103,185,180)(14,104,186,169)(15,105,187,170)(16,106,188,171)(17,107,189,172)(18,108,190,173)(19,97,191,174)(20,98,192,175)(21,99,181,176)(22,100,182,177)(23,101,183,178)(24,102,184,179)(25,138,88,43)(26,139,89,44)(27,140,90,45)(28,141,91,46)(29,142,92,47)(30,143,93,48)(31,144,94,37)(32,133,95,38)(33,134,96,39)(34,135,85,40)(35,136,86,41)(36,137,87,42)(49,146,122,164)(50,147,123,165)(51,148,124,166)(52,149,125,167)(53,150,126,168)(54,151,127,157)(55,152,128,158)(56,153,129,159)(57,154,130,160)(58,155,131,161)(59,156,132,162)(60,145,121,163), (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) );
G=PermutationGroup([(1,148),(2,149),(3,150),(4,151),(5,152),(6,153),(7,154),(8,155),(9,156),(10,145),(11,146),(12,147),(13,92),(14,93),(15,94),(16,95),(17,96),(18,85),(19,86),(20,87),(21,88),(22,89),(23,90),(24,91),(25,181),(26,182),(27,183),(28,184),(29,185),(30,186),(31,187),(32,188),(33,189),(34,190),(35,191),(36,192),(37,105),(38,106),(39,107),(40,108),(41,97),(42,98),(43,99),(44,100),(45,101),(46,102),(47,103),(48,104),(49,69),(50,70),(51,71),(52,72),(53,61),(54,62),(55,63),(56,64),(57,65),(58,66),(59,67),(60,68),(73,165),(74,166),(75,167),(76,168),(77,157),(78,158),(79,159),(80,160),(81,161),(82,162),(83,163),(84,164),(109,129),(110,130),(111,131),(112,132),(113,121),(114,122),(115,123),(116,124),(117,125),(118,126),(119,127),(120,128),(133,171),(134,172),(135,173),(136,174),(137,175),(138,176),(139,177),(140,178),(141,179),(142,180),(143,169),(144,170)], [(1,23),(2,24),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(11,21),(12,22),(25,164),(26,165),(27,166),(28,167),(29,168),(30,157),(31,158),(32,159),(33,160),(34,161),(35,162),(36,163),(37,128),(38,129),(39,130),(40,131),(41,132),(42,121),(43,122),(44,123),(45,124),(46,125),(47,126),(48,127),(49,138),(50,139),(51,140),(52,141),(53,142),(54,143),(55,144),(56,133),(57,134),(58,135),(59,136),(60,137),(61,180),(62,169),(63,170),(64,171),(65,172),(66,173),(67,174),(68,175),(69,176),(70,177),(71,178),(72,179),(73,182),(74,183),(75,184),(76,185),(77,186),(78,187),(79,188),(80,189),(81,190),(82,191),(83,192),(84,181),(85,155),(86,156),(87,145),(88,146),(89,147),(90,148),(91,149),(92,150),(93,151),(94,152),(95,153),(96,154),(97,112),(98,113),(99,114),(100,115),(101,116),(102,117),(103,118),(104,119),(105,120),(106,109),(107,110),(108,111)], [(1,116,74,71),(2,117,75,72),(3,118,76,61),(4,119,77,62),(5,120,78,63),(6,109,79,64),(7,110,80,65),(8,111,81,66),(9,112,82,67),(10,113,83,68),(11,114,84,69),(12,115,73,70),(13,103,185,180),(14,104,186,169),(15,105,187,170),(16,106,188,171),(17,107,189,172),(18,108,190,173),(19,97,191,174),(20,98,192,175),(21,99,181,176),(22,100,182,177),(23,101,183,178),(24,102,184,179),(25,138,88,43),(26,139,89,44),(27,140,90,45),(28,141,91,46),(29,142,92,47),(30,143,93,48),(31,144,94,37),(32,133,95,38),(33,134,96,39),(34,135,85,40),(35,136,86,41),(36,137,87,42),(49,146,122,164),(50,147,123,165),(51,148,124,166),(52,149,125,167),(53,150,126,168),(54,151,127,157),(55,152,128,158),(56,153,129,159),(57,154,130,160),(58,155,131,161),(59,156,132,162),(60,145,121,163)], [(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)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 12 |
| 3 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 5 |
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,8,0,0,0,0,1,0,0,0,0,12],[3,0,0,0,0,11,0,0,0,0,12,0,0,0,0,5] >;
192 conjugacy classes
| class | 1 | 2A | ··· | 2O | 3A | 3B | 4A | ··· | 4AV | 6A | ··· | 6AD | 12A | ··· | 12CR |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
192 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 |
| kernel | C22×C4×C12 | C2×C4×C12 | C23×C12 | C22×C42 | C22×C12 | C2×C42 | C23×C4 | C22×C4 |
| # reps | 1 | 12 | 3 | 2 | 48 | 24 | 6 | 96 |
In GAP, Magma, Sage, TeX
C_2^2\times C_4\times C_{12} % in TeX
G:=Group("C2^2xC4xC12"); // GroupNames label
G:=SmallGroup(192,1400);
// by ID
G=gap.SmallGroup(192,1400);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,336,680]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^4=d^12=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,c*d=d*c>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀