direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C22×D53, C53⋊C23, C106⋊C22, (C2×C106)⋊3C2, SmallGroup(424,13)
Series: Derived ►Chief ►Lower central ►Upper central
| C53 — C22×D53 |
Generators and relations for C22×D53
G = < a,b,c,d | a2=b2=c53=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Smallest permutation representation of C22×D53
►On 212 points
Generators in S212
(1 174)(2 175)(3 176)(4 177)(5 178)(6 179)(7 180)(8 181)(9 182)(10 183)(11 184)(12 185)(13 186)(14 187)(15 188)(16 189)(17 190)(18 191)(19 192)(20 193)(21 194)(22 195)(23 196)(24 197)(25 198)(26 199)(27 200)(28 201)(29 202)(30 203)(31 204)(32 205)(33 206)(34 207)(35 208)(36 209)(37 210)(38 211)(39 212)(40 160)(41 161)(42 162)(43 163)(44 164)(45 165)(46 166)(47 167)(48 168)(49 169)(50 170)(51 171)(52 172)(53 173)(54 157)(55 158)(56 159)(57 107)(58 108)(59 109)(60 110)(61 111)(62 112)(63 113)(64 114)(65 115)(66 116)(67 117)(68 118)(69 119)(70 120)(71 121)(72 122)(73 123)(74 124)(75 125)(76 126)(77 127)(78 128)(79 129)(80 130)(81 131)(82 132)(83 133)(84 134)(85 135)(86 136)(87 137)(88 138)(89 139)(90 140)(91 141)(92 142)(93 143)(94 144)(95 145)(96 146)(97 147)(98 148)(99 149)(100 150)(101 151)(102 152)(103 153)(104 154)(105 155)(106 156)
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 81)(10 82)(11 83)(12 84)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 97)(26 98)(27 99)(28 100)(29 101)(30 102)(31 103)(32 104)(33 105)(34 106)(35 54)(36 55)(37 56)(38 57)(39 58)(40 59)(41 60)(42 61)(43 62)(44 63)(45 64)(46 65)(47 66)(48 67)(49 68)(50 69)(51 70)(52 71)(53 72)(107 211)(108 212)(109 160)(110 161)(111 162)(112 163)(113 164)(114 165)(115 166)(116 167)(117 168)(118 169)(119 170)(120 171)(121 172)(122 173)(123 174)(124 175)(125 176)(126 177)(127 178)(128 179)(129 180)(130 181)(131 182)(132 183)(133 184)(134 185)(135 186)(136 187)(137 188)(138 189)(139 190)(140 191)(141 192)(142 193)(143 194)(144 195)(145 196)(146 197)(147 198)(148 199)(149 200)(150 201)(151 202)(152 203)(153 204)(154 205)(155 206)(156 207)(157 208)(158 209)(159 210)
(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)(160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212)
(1 122)(2 121)(3 120)(4 119)(5 118)(6 117)(7 116)(8 115)(9 114)(10 113)(11 112)(12 111)(13 110)(14 109)(15 108)(16 107)(17 159)(18 158)(19 157)(20 156)(21 155)(22 154)(23 153)(24 152)(25 151)(26 150)(27 149)(28 148)(29 147)(30 146)(31 145)(32 144)(33 143)(34 142)(35 141)(36 140)(37 139)(38 138)(39 137)(40 136)(41 135)(42 134)(43 133)(44 132)(45 131)(46 130)(47 129)(48 128)(49 127)(50 126)(51 125)(52 124)(53 123)(54 192)(55 191)(56 190)(57 189)(58 188)(59 187)(60 186)(61 185)(62 184)(63 183)(64 182)(65 181)(66 180)(67 179)(68 178)(69 177)(70 176)(71 175)(72 174)(73 173)(74 172)(75 171)(76 170)(77 169)(78 168)(79 167)(80 166)(81 165)(82 164)(83 163)(84 162)(85 161)(86 160)(87 212)(88 211)(89 210)(90 209)(91 208)(92 207)(93 206)(94 205)(95 204)(96 203)(97 202)(98 201)(99 200)(100 199)(101 198)(102 197)(103 196)(104 195)(105 194)(106 193)
G:=sub<Sym(212)| (1,174)(2,175)(3,176)(4,177)(5,178)(6,179)(7,180)(8,181)(9,182)(10,183)(11,184)(12,185)(13,186)(14,187)(15,188)(16,189)(17,190)(18,191)(19,192)(20,193)(21,194)(22,195)(23,196)(24,197)(25,198)(26,199)(27,200)(28,201)(29,202)(30,203)(31,204)(32,205)(33,206)(34,207)(35,208)(36,209)(37,210)(38,211)(39,212)(40,160)(41,161)(42,162)(43,163)(44,164)(45,165)(46,166)(47,167)(48,168)(49,169)(50,170)(51,171)(52,172)(53,173)(54,157)(55,158)(56,159)(57,107)(58,108)(59,109)(60,110)(61,111)(62,112)(63,113)(64,114)(65,115)(66,116)(67,117)(68,118)(69,119)(70,120)(71,121)(72,122)(73,123)(74,124)(75,125)(76,126)(77,127)(78,128)(79,129)(80,130)(81,131)(82,132)(83,133)(84,134)(85,135)(86,136)(87,137)(88,138)(89,139)(90,140)(91,141)(92,142)(93,143)(94,144)(95,145)(96,146)(97,147)(98,148)(99,149)(100,150)(101,151)(102,152)(103,153)(104,154)(105,155)(106,156), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,81)(10,82)(11,83)(12,84)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,97)(26,98)(27,99)(28,100)(29,101)(30,102)(31,103)(32,104)(33,105)(34,106)(35,54)(36,55)(37,56)(38,57)(39,58)(40,59)(41,60)(42,61)(43,62)(44,63)(45,64)(46,65)(47,66)(48,67)(49,68)(50,69)(51,70)(52,71)(53,72)(107,211)(108,212)(109,160)(110,161)(111,162)(112,163)(113,164)(114,165)(115,166)(116,167)(117,168)(118,169)(119,170)(120,171)(121,172)(122,173)(123,174)(124,175)(125,176)(126,177)(127,178)(128,179)(129,180)(130,181)(131,182)(132,183)(133,184)(134,185)(135,186)(136,187)(137,188)(138,189)(139,190)(140,191)(141,192)(142,193)(143,194)(144,195)(145,196)(146,197)(147,198)(148,199)(149,200)(150,201)(151,202)(152,203)(153,204)(154,205)(155,206)(156,207)(157,208)(158,209)(159,210), (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)(160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212), (1,122)(2,121)(3,120)(4,119)(5,118)(6,117)(7,116)(8,115)(9,114)(10,113)(11,112)(12,111)(13,110)(14,109)(15,108)(16,107)(17,159)(18,158)(19,157)(20,156)(21,155)(22,154)(23,153)(24,152)(25,151)(26,150)(27,149)(28,148)(29,147)(30,146)(31,145)(32,144)(33,143)(34,142)(35,141)(36,140)(37,139)(38,138)(39,137)(40,136)(41,135)(42,134)(43,133)(44,132)(45,131)(46,130)(47,129)(48,128)(49,127)(50,126)(51,125)(52,124)(53,123)(54,192)(55,191)(56,190)(57,189)(58,188)(59,187)(60,186)(61,185)(62,184)(63,183)(64,182)(65,181)(66,180)(67,179)(68,178)(69,177)(70,176)(71,175)(72,174)(73,173)(74,172)(75,171)(76,170)(77,169)(78,168)(79,167)(80,166)(81,165)(82,164)(83,163)(84,162)(85,161)(86,160)(87,212)(88,211)(89,210)(90,209)(91,208)(92,207)(93,206)(94,205)(95,204)(96,203)(97,202)(98,201)(99,200)(100,199)(101,198)(102,197)(103,196)(104,195)(105,194)(106,193)>;
G:=Group( (1,174)(2,175)(3,176)(4,177)(5,178)(6,179)(7,180)(8,181)(9,182)(10,183)(11,184)(12,185)(13,186)(14,187)(15,188)(16,189)(17,190)(18,191)(19,192)(20,193)(21,194)(22,195)(23,196)(24,197)(25,198)(26,199)(27,200)(28,201)(29,202)(30,203)(31,204)(32,205)(33,206)(34,207)(35,208)(36,209)(37,210)(38,211)(39,212)(40,160)(41,161)(42,162)(43,163)(44,164)(45,165)(46,166)(47,167)(48,168)(49,169)(50,170)(51,171)(52,172)(53,173)(54,157)(55,158)(56,159)(57,107)(58,108)(59,109)(60,110)(61,111)(62,112)(63,113)(64,114)(65,115)(66,116)(67,117)(68,118)(69,119)(70,120)(71,121)(72,122)(73,123)(74,124)(75,125)(76,126)(77,127)(78,128)(79,129)(80,130)(81,131)(82,132)(83,133)(84,134)(85,135)(86,136)(87,137)(88,138)(89,139)(90,140)(91,141)(92,142)(93,143)(94,144)(95,145)(96,146)(97,147)(98,148)(99,149)(100,150)(101,151)(102,152)(103,153)(104,154)(105,155)(106,156), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,81)(10,82)(11,83)(12,84)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,97)(26,98)(27,99)(28,100)(29,101)(30,102)(31,103)(32,104)(33,105)(34,106)(35,54)(36,55)(37,56)(38,57)(39,58)(40,59)(41,60)(42,61)(43,62)(44,63)(45,64)(46,65)(47,66)(48,67)(49,68)(50,69)(51,70)(52,71)(53,72)(107,211)(108,212)(109,160)(110,161)(111,162)(112,163)(113,164)(114,165)(115,166)(116,167)(117,168)(118,169)(119,170)(120,171)(121,172)(122,173)(123,174)(124,175)(125,176)(126,177)(127,178)(128,179)(129,180)(130,181)(131,182)(132,183)(133,184)(134,185)(135,186)(136,187)(137,188)(138,189)(139,190)(140,191)(141,192)(142,193)(143,194)(144,195)(145,196)(146,197)(147,198)(148,199)(149,200)(150,201)(151,202)(152,203)(153,204)(154,205)(155,206)(156,207)(157,208)(158,209)(159,210), (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)(160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212), (1,122)(2,121)(3,120)(4,119)(5,118)(6,117)(7,116)(8,115)(9,114)(10,113)(11,112)(12,111)(13,110)(14,109)(15,108)(16,107)(17,159)(18,158)(19,157)(20,156)(21,155)(22,154)(23,153)(24,152)(25,151)(26,150)(27,149)(28,148)(29,147)(30,146)(31,145)(32,144)(33,143)(34,142)(35,141)(36,140)(37,139)(38,138)(39,137)(40,136)(41,135)(42,134)(43,133)(44,132)(45,131)(46,130)(47,129)(48,128)(49,127)(50,126)(51,125)(52,124)(53,123)(54,192)(55,191)(56,190)(57,189)(58,188)(59,187)(60,186)(61,185)(62,184)(63,183)(64,182)(65,181)(66,180)(67,179)(68,178)(69,177)(70,176)(71,175)(72,174)(73,173)(74,172)(75,171)(76,170)(77,169)(78,168)(79,167)(80,166)(81,165)(82,164)(83,163)(84,162)(85,161)(86,160)(87,212)(88,211)(89,210)(90,209)(91,208)(92,207)(93,206)(94,205)(95,204)(96,203)(97,202)(98,201)(99,200)(100,199)(101,198)(102,197)(103,196)(104,195)(105,194)(106,193) );
G=PermutationGroup([[(1,174),(2,175),(3,176),(4,177),(5,178),(6,179),(7,180),(8,181),(9,182),(10,183),(11,184),(12,185),(13,186),(14,187),(15,188),(16,189),(17,190),(18,191),(19,192),(20,193),(21,194),(22,195),(23,196),(24,197),(25,198),(26,199),(27,200),(28,201),(29,202),(30,203),(31,204),(32,205),(33,206),(34,207),(35,208),(36,209),(37,210),(38,211),(39,212),(40,160),(41,161),(42,162),(43,163),(44,164),(45,165),(46,166),(47,167),(48,168),(49,169),(50,170),(51,171),(52,172),(53,173),(54,157),(55,158),(56,159),(57,107),(58,108),(59,109),(60,110),(61,111),(62,112),(63,113),(64,114),(65,115),(66,116),(67,117),(68,118),(69,119),(70,120),(71,121),(72,122),(73,123),(74,124),(75,125),(76,126),(77,127),(78,128),(79,129),(80,130),(81,131),(82,132),(83,133),(84,134),(85,135),(86,136),(87,137),(88,138),(89,139),(90,140),(91,141),(92,142),(93,143),(94,144),(95,145),(96,146),(97,147),(98,148),(99,149),(100,150),(101,151),(102,152),(103,153),(104,154),(105,155),(106,156)], [(1,73),(2,74),(3,75),(4,76),(5,77),(6,78),(7,79),(8,80),(9,81),(10,82),(11,83),(12,84),(13,85),(14,86),(15,87),(16,88),(17,89),(18,90),(19,91),(20,92),(21,93),(22,94),(23,95),(24,96),(25,97),(26,98),(27,99),(28,100),(29,101),(30,102),(31,103),(32,104),(33,105),(34,106),(35,54),(36,55),(37,56),(38,57),(39,58),(40,59),(41,60),(42,61),(43,62),(44,63),(45,64),(46,65),(47,66),(48,67),(49,68),(50,69),(51,70),(52,71),(53,72),(107,211),(108,212),(109,160),(110,161),(111,162),(112,163),(113,164),(114,165),(115,166),(116,167),(117,168),(118,169),(119,170),(120,171),(121,172),(122,173),(123,174),(124,175),(125,176),(126,177),(127,178),(128,179),(129,180),(130,181),(131,182),(132,183),(133,184),(134,185),(135,186),(136,187),(137,188),(138,189),(139,190),(140,191),(141,192),(142,193),(143,194),(144,195),(145,196),(146,197),(147,198),(148,199),(149,200),(150,201),(151,202),(152,203),(153,204),(154,205),(155,206),(156,207),(157,208),(158,209),(159,210)], [(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),(160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212)], [(1,122),(2,121),(3,120),(4,119),(5,118),(6,117),(7,116),(8,115),(9,114),(10,113),(11,112),(12,111),(13,110),(14,109),(15,108),(16,107),(17,159),(18,158),(19,157),(20,156),(21,155),(22,154),(23,153),(24,152),(25,151),(26,150),(27,149),(28,148),(29,147),(30,146),(31,145),(32,144),(33,143),(34,142),(35,141),(36,140),(37,139),(38,138),(39,137),(40,136),(41,135),(42,134),(43,133),(44,132),(45,131),(46,130),(47,129),(48,128),(49,127),(50,126),(51,125),(52,124),(53,123),(54,192),(55,191),(56,190),(57,189),(58,188),(59,187),(60,186),(61,185),(62,184),(63,183),(64,182),(65,181),(66,180),(67,179),(68,178),(69,177),(70,176),(71,175),(72,174),(73,173),(74,172),(75,171),(76,170),(77,169),(78,168),(79,167),(80,166),(81,165),(82,164),(83,163),(84,162),(85,161),(86,160),(87,212),(88,211),(89,210),(90,209),(91,208),(92,207),(93,206),(94,205),(95,204),(96,203),(97,202),(98,201),(99,200),(100,199),(101,198),(102,197),(103,196),(104,195),(105,194),(106,193)]])
112 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 53A | ··· | 53Z | 106A | ··· | 106BZ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 53 | ··· | 53 | 106 | ··· | 106 |
| size | 1 | 1 | 1 | 1 | 53 | 53 | 53 | 53 | 2 | ··· | 2 | 2 | ··· | 2 |
112 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | C2 | D53 | D106 |
| kernel | C22×D53 | D106 | C2×C106 | C22 | C2 |
| # reps | 1 | 6 | 1 | 26 | 78 |
Matrix representation of C22×D53 ►in GL4(𝔽107) generated by
| 1 | 0 | 0 | 0 |
| 0 | 106 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 106 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 36 | 1 |
| 0 | 0 | 86 | 47 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 37 | 15 |
| 0 | 0 | 80 | 70 |
G:=sub<GL(4,GF(107))| [1,0,0,0,0,106,0,0,0,0,1,0,0,0,0,1],[106,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,36,86,0,0,1,47],[1,0,0,0,0,1,0,0,0,0,37,80,0,0,15,70] >;
C22×D53 in GAP, Magma, Sage, TeX
C_2^2\times D_{53} % in TeX
G:=Group("C2^2xD53"); // GroupNames label
G:=SmallGroup(424,13);
// by ID
G=gap.SmallGroup(424,13);
# by ID
G:=PCGroup([4,-2,-2,-2,-53,6659]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^53=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
Export