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 |
Generators and relations for C22×C4×C12
G = < a,b,c,d | a2=b2=c4=d12=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >
Subgroups: 498, all normal (8 characteristic)
C1, C2, C3, C4, C22, C22, C6, C2×C4, C23, C12, C2×C6, C2×C6, C42, C22×C4, C24, C2×C12, C22×C6, C2×C42, C23×C4, C4×C12, C22×C12, C23×C6, C22×C42, C2×C4×C12, C23×C12, C22×C4×C12
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, C42, C22×C4, C24, C2×C12, C22×C6, C2×C42, C23×C4, C4×C12, C22×C12, C23×C6, C22×C42, C2×C4×C12, C23×C12, C22×C4×C12
►Regular action on 192 points
Generators in S192
(1 192)(2 181)(3 182)(4 183)(5 184)(6 185)(7 186)(8 187)(9 188)(10 189)(11 190)(12 191)(13 91)(14 92)(15 93)(16 94)(17 95)(18 96)(19 85)(20 86)(21 87)(22 88)(23 89)(24 90)(25 46)(26 47)(27 48)(28 37)(29 38)(30 39)(31 40)(32 41)(33 42)(34 43)(35 44)(36 45)(49 122)(50 123)(51 124)(52 125)(53 126)(54 127)(55 128)(56 129)(57 130)(58 131)(59 132)(60 121)(61 137)(62 138)(63 139)(64 140)(65 141)(66 142)(67 143)(68 144)(69 133)(70 134)(71 135)(72 136)(73 164)(74 165)(75 166)(76 167)(77 168)(78 157)(79 158)(80 159)(81 160)(82 161)(83 162)(84 163)(97 153)(98 154)(99 155)(100 156)(101 145)(102 146)(103 147)(104 148)(105 149)(106 150)(107 151)(108 152)(109 170)(110 171)(111 172)(112 173)(113 174)(114 175)(115 176)(116 177)(117 178)(118 179)(119 180)(120 169)
(1 109)(2 110)(3 111)(4 112)(5 113)(6 114)(7 115)(8 116)(9 117)(10 118)(11 119)(12 120)(13 148)(14 149)(15 150)(16 151)(17 152)(18 153)(19 154)(20 155)(21 156)(22 145)(23 146)(24 147)(25 137)(26 138)(27 139)(28 140)(29 141)(30 142)(31 143)(32 144)(33 133)(34 134)(35 135)(36 136)(37 64)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(46 61)(47 62)(48 63)(49 165)(50 166)(51 167)(52 168)(53 157)(54 158)(55 159)(56 160)(57 161)(58 162)(59 163)(60 164)(73 121)(74 122)(75 123)(76 124)(77 125)(78 126)(79 127)(80 128)(81 129)(82 130)(83 131)(84 132)(85 98)(86 99)(87 100)(88 101)(89 102)(90 103)(91 104)(92 105)(93 106)(94 107)(95 108)(96 97)(169 191)(170 192)(171 181)(172 182)(173 183)(174 184)(175 185)(176 186)(177 187)(178 188)(179 189)(180 190)
(1 84 68 102)(2 73 69 103)(3 74 70 104)(4 75 71 105)(5 76 72 106)(6 77 61 107)(7 78 62 108)(8 79 63 97)(9 80 64 98)(10 81 65 99)(11 82 66 100)(12 83 67 101)(13 172 49 34)(14 173 50 35)(15 174 51 36)(16 175 52 25)(17 176 53 26)(18 177 54 27)(19 178 55 28)(20 179 56 29)(21 180 57 30)(22 169 58 31)(23 170 59 32)(24 171 60 33)(37 85 117 128)(38 86 118 129)(39 87 119 130)(40 88 120 131)(41 89 109 132)(42 90 110 121)(43 91 111 122)(44 92 112 123)(45 93 113 124)(46 94 114 125)(47 95 115 126)(48 96 116 127)(133 147 181 164)(134 148 182 165)(135 149 183 166)(136 150 184 167)(137 151 185 168)(138 152 186 157)(139 153 187 158)(140 154 188 159)(141 155 189 160)(142 156 190 161)(143 145 191 162)(144 146 192 163)
(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)| (1,192)(2,181)(3,182)(4,183)(5,184)(6,185)(7,186)(8,187)(9,188)(10,189)(11,190)(12,191)(13,91)(14,92)(15,93)(16,94)(17,95)(18,96)(19,85)(20,86)(21,87)(22,88)(23,89)(24,90)(25,46)(26,47)(27,48)(28,37)(29,38)(30,39)(31,40)(32,41)(33,42)(34,43)(35,44)(36,45)(49,122)(50,123)(51,124)(52,125)(53,126)(54,127)(55,128)(56,129)(57,130)(58,131)(59,132)(60,121)(61,137)(62,138)(63,139)(64,140)(65,141)(66,142)(67,143)(68,144)(69,133)(70,134)(71,135)(72,136)(73,164)(74,165)(75,166)(76,167)(77,168)(78,157)(79,158)(80,159)(81,160)(82,161)(83,162)(84,163)(97,153)(98,154)(99,155)(100,156)(101,145)(102,146)(103,147)(104,148)(105,149)(106,150)(107,151)(108,152)(109,170)(110,171)(111,172)(112,173)(113,174)(114,175)(115,176)(116,177)(117,178)(118,179)(119,180)(120,169), (1,109)(2,110)(3,111)(4,112)(5,113)(6,114)(7,115)(8,116)(9,117)(10,118)(11,119)(12,120)(13,148)(14,149)(15,150)(16,151)(17,152)(18,153)(19,154)(20,155)(21,156)(22,145)(23,146)(24,147)(25,137)(26,138)(27,139)(28,140)(29,141)(30,142)(31,143)(32,144)(33,133)(34,134)(35,135)(36,136)(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,61)(47,62)(48,63)(49,165)(50,166)(51,167)(52,168)(53,157)(54,158)(55,159)(56,160)(57,161)(58,162)(59,163)(60,164)(73,121)(74,122)(75,123)(76,124)(77,125)(78,126)(79,127)(80,128)(81,129)(82,130)(83,131)(84,132)(85,98)(86,99)(87,100)(88,101)(89,102)(90,103)(91,104)(92,105)(93,106)(94,107)(95,108)(96,97)(169,191)(170,192)(171,181)(172,182)(173,183)(174,184)(175,185)(176,186)(177,187)(178,188)(179,189)(180,190), (1,84,68,102)(2,73,69,103)(3,74,70,104)(4,75,71,105)(5,76,72,106)(6,77,61,107)(7,78,62,108)(8,79,63,97)(9,80,64,98)(10,81,65,99)(11,82,66,100)(12,83,67,101)(13,172,49,34)(14,173,50,35)(15,174,51,36)(16,175,52,25)(17,176,53,26)(18,177,54,27)(19,178,55,28)(20,179,56,29)(21,180,57,30)(22,169,58,31)(23,170,59,32)(24,171,60,33)(37,85,117,128)(38,86,118,129)(39,87,119,130)(40,88,120,131)(41,89,109,132)(42,90,110,121)(43,91,111,122)(44,92,112,123)(45,93,113,124)(46,94,114,125)(47,95,115,126)(48,96,116,127)(133,147,181,164)(134,148,182,165)(135,149,183,166)(136,150,184,167)(137,151,185,168)(138,152,186,157)(139,153,187,158)(140,154,188,159)(141,155,189,160)(142,156,190,161)(143,145,191,162)(144,146,192,163), (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( (1,192)(2,181)(3,182)(4,183)(5,184)(6,185)(7,186)(8,187)(9,188)(10,189)(11,190)(12,191)(13,91)(14,92)(15,93)(16,94)(17,95)(18,96)(19,85)(20,86)(21,87)(22,88)(23,89)(24,90)(25,46)(26,47)(27,48)(28,37)(29,38)(30,39)(31,40)(32,41)(33,42)(34,43)(35,44)(36,45)(49,122)(50,123)(51,124)(52,125)(53,126)(54,127)(55,128)(56,129)(57,130)(58,131)(59,132)(60,121)(61,137)(62,138)(63,139)(64,140)(65,141)(66,142)(67,143)(68,144)(69,133)(70,134)(71,135)(72,136)(73,164)(74,165)(75,166)(76,167)(77,168)(78,157)(79,158)(80,159)(81,160)(82,161)(83,162)(84,163)(97,153)(98,154)(99,155)(100,156)(101,145)(102,146)(103,147)(104,148)(105,149)(106,150)(107,151)(108,152)(109,170)(110,171)(111,172)(112,173)(113,174)(114,175)(115,176)(116,177)(117,178)(118,179)(119,180)(120,169), (1,109)(2,110)(3,111)(4,112)(5,113)(6,114)(7,115)(8,116)(9,117)(10,118)(11,119)(12,120)(13,148)(14,149)(15,150)(16,151)(17,152)(18,153)(19,154)(20,155)(21,156)(22,145)(23,146)(24,147)(25,137)(26,138)(27,139)(28,140)(29,141)(30,142)(31,143)(32,144)(33,133)(34,134)(35,135)(36,136)(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,61)(47,62)(48,63)(49,165)(50,166)(51,167)(52,168)(53,157)(54,158)(55,159)(56,160)(57,161)(58,162)(59,163)(60,164)(73,121)(74,122)(75,123)(76,124)(77,125)(78,126)(79,127)(80,128)(81,129)(82,130)(83,131)(84,132)(85,98)(86,99)(87,100)(88,101)(89,102)(90,103)(91,104)(92,105)(93,106)(94,107)(95,108)(96,97)(169,191)(170,192)(171,181)(172,182)(173,183)(174,184)(175,185)(176,186)(177,187)(178,188)(179,189)(180,190), (1,84,68,102)(2,73,69,103)(3,74,70,104)(4,75,71,105)(5,76,72,106)(6,77,61,107)(7,78,62,108)(8,79,63,97)(9,80,64,98)(10,81,65,99)(11,82,66,100)(12,83,67,101)(13,172,49,34)(14,173,50,35)(15,174,51,36)(16,175,52,25)(17,176,53,26)(18,177,54,27)(19,178,55,28)(20,179,56,29)(21,180,57,30)(22,169,58,31)(23,170,59,32)(24,171,60,33)(37,85,117,128)(38,86,118,129)(39,87,119,130)(40,88,120,131)(41,89,109,132)(42,90,110,121)(43,91,111,122)(44,92,112,123)(45,93,113,124)(46,94,114,125)(47,95,115,126)(48,96,116,127)(133,147,181,164)(134,148,182,165)(135,149,183,166)(136,150,184,167)(137,151,185,168)(138,152,186,157)(139,153,187,158)(140,154,188,159)(141,155,189,160)(142,156,190,161)(143,145,191,162)(144,146,192,163), (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([[(1,192),(2,181),(3,182),(4,183),(5,184),(6,185),(7,186),(8,187),(9,188),(10,189),(11,190),(12,191),(13,91),(14,92),(15,93),(16,94),(17,95),(18,96),(19,85),(20,86),(21,87),(22,88),(23,89),(24,90),(25,46),(26,47),(27,48),(28,37),(29,38),(30,39),(31,40),(32,41),(33,42),(34,43),(35,44),(36,45),(49,122),(50,123),(51,124),(52,125),(53,126),(54,127),(55,128),(56,129),(57,130),(58,131),(59,132),(60,121),(61,137),(62,138),(63,139),(64,140),(65,141),(66,142),(67,143),(68,144),(69,133),(70,134),(71,135),(72,136),(73,164),(74,165),(75,166),(76,167),(77,168),(78,157),(79,158),(80,159),(81,160),(82,161),(83,162),(84,163),(97,153),(98,154),(99,155),(100,156),(101,145),(102,146),(103,147),(104,148),(105,149),(106,150),(107,151),(108,152),(109,170),(110,171),(111,172),(112,173),(113,174),(114,175),(115,176),(116,177),(117,178),(118,179),(119,180),(120,169)], [(1,109),(2,110),(3,111),(4,112),(5,113),(6,114),(7,115),(8,116),(9,117),(10,118),(11,119),(12,120),(13,148),(14,149),(15,150),(16,151),(17,152),(18,153),(19,154),(20,155),(21,156),(22,145),(23,146),(24,147),(25,137),(26,138),(27,139),(28,140),(29,141),(30,142),(31,143),(32,144),(33,133),(34,134),(35,135),(36,136),(37,64),(38,65),(39,66),(40,67),(41,68),(42,69),(43,70),(44,71),(45,72),(46,61),(47,62),(48,63),(49,165),(50,166),(51,167),(52,168),(53,157),(54,158),(55,159),(56,160),(57,161),(58,162),(59,163),(60,164),(73,121),(74,122),(75,123),(76,124),(77,125),(78,126),(79,127),(80,128),(81,129),(82,130),(83,131),(84,132),(85,98),(86,99),(87,100),(88,101),(89,102),(90,103),(91,104),(92,105),(93,106),(94,107),(95,108),(96,97),(169,191),(170,192),(171,181),(172,182),(173,183),(174,184),(175,185),(176,186),(177,187),(178,188),(179,189),(180,190)], [(1,84,68,102),(2,73,69,103),(3,74,70,104),(4,75,71,105),(5,76,72,106),(6,77,61,107),(7,78,62,108),(8,79,63,97),(9,80,64,98),(10,81,65,99),(11,82,66,100),(12,83,67,101),(13,172,49,34),(14,173,50,35),(15,174,51,36),(16,175,52,25),(17,176,53,26),(18,177,54,27),(19,178,55,28),(20,179,56,29),(21,180,57,30),(22,169,58,31),(23,170,59,32),(24,171,60,33),(37,85,117,128),(38,86,118,129),(39,87,119,130),(40,88,120,131),(41,89,109,132),(42,90,110,121),(43,91,111,122),(44,92,112,123),(45,93,113,124),(46,94,114,125),(47,95,115,126),(48,96,116,127),(133,147,181,164),(134,148,182,165),(135,149,183,166),(136,150,184,167),(137,151,185,168),(138,152,186,157),(139,153,187,158),(140,154,188,159),(141,155,189,160),(142,156,190,161),(143,145,191,162),(144,146,192,163)], [(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)]])
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 |
Matrix representation of C22×C4×C12 ►in 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] >;
C22×C4×C12 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