direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×D53, C212⋊2C2, C2.1D106, Dic53⋊2C2, D106.2C2, C106.2C22, C53⋊2(C2×C4), SmallGroup(424,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C53 — C4×D53 |
Generators and relations for C4×D53
G = < a,b,c | a4=b53=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C4×D53
►On 212 points
Generators in S212
(1 184 102 157)(2 185 103 158)(3 186 104 159)(4 187 105 107)(5 188 106 108)(6 189 54 109)(7 190 55 110)(8 191 56 111)(9 192 57 112)(10 193 58 113)(11 194 59 114)(12 195 60 115)(13 196 61 116)(14 197 62 117)(15 198 63 118)(16 199 64 119)(17 200 65 120)(18 201 66 121)(19 202 67 122)(20 203 68 123)(21 204 69 124)(22 205 70 125)(23 206 71 126)(24 207 72 127)(25 208 73 128)(26 209 74 129)(27 210 75 130)(28 211 76 131)(29 212 77 132)(30 160 78 133)(31 161 79 134)(32 162 80 135)(33 163 81 136)(34 164 82 137)(35 165 83 138)(36 166 84 139)(37 167 85 140)(38 168 86 141)(39 169 87 142)(40 170 88 143)(41 171 89 144)(42 172 90 145)(43 173 91 146)(44 174 92 147)(45 175 93 148)(46 176 94 149)(47 177 95 150)(48 178 96 151)(49 179 97 152)(50 180 98 153)(51 181 99 154)(52 182 100 155)(53 183 101 156)
(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 53)(2 52)(3 51)(4 50)(5 49)(6 48)(7 47)(8 46)(9 45)(10 44)(11 43)(12 42)(13 41)(14 40)(15 39)(16 38)(17 37)(18 36)(19 35)(20 34)(21 33)(22 32)(23 31)(24 30)(25 29)(26 28)(54 96)(55 95)(56 94)(57 93)(58 92)(59 91)(60 90)(61 89)(62 88)(63 87)(64 86)(65 85)(66 84)(67 83)(68 82)(69 81)(70 80)(71 79)(72 78)(73 77)(74 76)(97 106)(98 105)(99 104)(100 103)(101 102)(107 153)(108 152)(109 151)(110 150)(111 149)(112 148)(113 147)(114 146)(115 145)(116 144)(117 143)(118 142)(119 141)(120 140)(121 139)(122 138)(123 137)(124 136)(125 135)(126 134)(127 133)(128 132)(129 131)(154 159)(155 158)(156 157)(160 207)(161 206)(162 205)(163 204)(164 203)(165 202)(166 201)(167 200)(168 199)(169 198)(170 197)(171 196)(172 195)(173 194)(174 193)(175 192)(176 191)(177 190)(178 189)(179 188)(180 187)(181 186)(182 185)(183 184)(208 212)(209 211)
G:=sub<Sym(212)| (1,184,102,157)(2,185,103,158)(3,186,104,159)(4,187,105,107)(5,188,106,108)(6,189,54,109)(7,190,55,110)(8,191,56,111)(9,192,57,112)(10,193,58,113)(11,194,59,114)(12,195,60,115)(13,196,61,116)(14,197,62,117)(15,198,63,118)(16,199,64,119)(17,200,65,120)(18,201,66,121)(19,202,67,122)(20,203,68,123)(21,204,69,124)(22,205,70,125)(23,206,71,126)(24,207,72,127)(25,208,73,128)(26,209,74,129)(27,210,75,130)(28,211,76,131)(29,212,77,132)(30,160,78,133)(31,161,79,134)(32,162,80,135)(33,163,81,136)(34,164,82,137)(35,165,83,138)(36,166,84,139)(37,167,85,140)(38,168,86,141)(39,169,87,142)(40,170,88,143)(41,171,89,144)(42,172,90,145)(43,173,91,146)(44,174,92,147)(45,175,93,148)(46,176,94,149)(47,177,95,150)(48,178,96,151)(49,179,97,152)(50,180,98,153)(51,181,99,154)(52,182,100,155)(53,183,101,156), (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), (1,53)(2,52)(3,51)(4,50)(5,49)(6,48)(7,47)(8,46)(9,45)(10,44)(11,43)(12,42)(13,41)(14,40)(15,39)(16,38)(17,37)(18,36)(19,35)(20,34)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(54,96)(55,95)(56,94)(57,93)(58,92)(59,91)(60,90)(61,89)(62,88)(63,87)(64,86)(65,85)(66,84)(67,83)(68,82)(69,81)(70,80)(71,79)(72,78)(73,77)(74,76)(97,106)(98,105)(99,104)(100,103)(101,102)(107,153)(108,152)(109,151)(110,150)(111,149)(112,148)(113,147)(114,146)(115,145)(116,144)(117,143)(118,142)(119,141)(120,140)(121,139)(122,138)(123,137)(124,136)(125,135)(126,134)(127,133)(128,132)(129,131)(154,159)(155,158)(156,157)(160,207)(161,206)(162,205)(163,204)(164,203)(165,202)(166,201)(167,200)(168,199)(169,198)(170,197)(171,196)(172,195)(173,194)(174,193)(175,192)(176,191)(177,190)(178,189)(179,188)(180,187)(181,186)(182,185)(183,184)(208,212)(209,211)>;
G:=Group( (1,184,102,157)(2,185,103,158)(3,186,104,159)(4,187,105,107)(5,188,106,108)(6,189,54,109)(7,190,55,110)(8,191,56,111)(9,192,57,112)(10,193,58,113)(11,194,59,114)(12,195,60,115)(13,196,61,116)(14,197,62,117)(15,198,63,118)(16,199,64,119)(17,200,65,120)(18,201,66,121)(19,202,67,122)(20,203,68,123)(21,204,69,124)(22,205,70,125)(23,206,71,126)(24,207,72,127)(25,208,73,128)(26,209,74,129)(27,210,75,130)(28,211,76,131)(29,212,77,132)(30,160,78,133)(31,161,79,134)(32,162,80,135)(33,163,81,136)(34,164,82,137)(35,165,83,138)(36,166,84,139)(37,167,85,140)(38,168,86,141)(39,169,87,142)(40,170,88,143)(41,171,89,144)(42,172,90,145)(43,173,91,146)(44,174,92,147)(45,175,93,148)(46,176,94,149)(47,177,95,150)(48,178,96,151)(49,179,97,152)(50,180,98,153)(51,181,99,154)(52,182,100,155)(53,183,101,156), (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), (1,53)(2,52)(3,51)(4,50)(5,49)(6,48)(7,47)(8,46)(9,45)(10,44)(11,43)(12,42)(13,41)(14,40)(15,39)(16,38)(17,37)(18,36)(19,35)(20,34)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(54,96)(55,95)(56,94)(57,93)(58,92)(59,91)(60,90)(61,89)(62,88)(63,87)(64,86)(65,85)(66,84)(67,83)(68,82)(69,81)(70,80)(71,79)(72,78)(73,77)(74,76)(97,106)(98,105)(99,104)(100,103)(101,102)(107,153)(108,152)(109,151)(110,150)(111,149)(112,148)(113,147)(114,146)(115,145)(116,144)(117,143)(118,142)(119,141)(120,140)(121,139)(122,138)(123,137)(124,136)(125,135)(126,134)(127,133)(128,132)(129,131)(154,159)(155,158)(156,157)(160,207)(161,206)(162,205)(163,204)(164,203)(165,202)(166,201)(167,200)(168,199)(169,198)(170,197)(171,196)(172,195)(173,194)(174,193)(175,192)(176,191)(177,190)(178,189)(179,188)(180,187)(181,186)(182,185)(183,184)(208,212)(209,211) );
G=PermutationGroup([[(1,184,102,157),(2,185,103,158),(3,186,104,159),(4,187,105,107),(5,188,106,108),(6,189,54,109),(7,190,55,110),(8,191,56,111),(9,192,57,112),(10,193,58,113),(11,194,59,114),(12,195,60,115),(13,196,61,116),(14,197,62,117),(15,198,63,118),(16,199,64,119),(17,200,65,120),(18,201,66,121),(19,202,67,122),(20,203,68,123),(21,204,69,124),(22,205,70,125),(23,206,71,126),(24,207,72,127),(25,208,73,128),(26,209,74,129),(27,210,75,130),(28,211,76,131),(29,212,77,132),(30,160,78,133),(31,161,79,134),(32,162,80,135),(33,163,81,136),(34,164,82,137),(35,165,83,138),(36,166,84,139),(37,167,85,140),(38,168,86,141),(39,169,87,142),(40,170,88,143),(41,171,89,144),(42,172,90,145),(43,173,91,146),(44,174,92,147),(45,175,93,148),(46,176,94,149),(47,177,95,150),(48,178,96,151),(49,179,97,152),(50,180,98,153),(51,181,99,154),(52,182,100,155),(53,183,101,156)], [(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)], [(1,53),(2,52),(3,51),(4,50),(5,49),(6,48),(7,47),(8,46),(9,45),(10,44),(11,43),(12,42),(13,41),(14,40),(15,39),(16,38),(17,37),(18,36),(19,35),(20,34),(21,33),(22,32),(23,31),(24,30),(25,29),(26,28),(54,96),(55,95),(56,94),(57,93),(58,92),(59,91),(60,90),(61,89),(62,88),(63,87),(64,86),(65,85),(66,84),(67,83),(68,82),(69,81),(70,80),(71,79),(72,78),(73,77),(74,76),(97,106),(98,105),(99,104),(100,103),(101,102),(107,153),(108,152),(109,151),(110,150),(111,149),(112,148),(113,147),(114,146),(115,145),(116,144),(117,143),(118,142),(119,141),(120,140),(121,139),(122,138),(123,137),(124,136),(125,135),(126,134),(127,133),(128,132),(129,131),(154,159),(155,158),(156,157),(160,207),(161,206),(162,205),(163,204),(164,203),(165,202),(166,201),(167,200),(168,199),(169,198),(170,197),(171,196),(172,195),(173,194),(174,193),(175,192),(176,191),(177,190),(178,189),(179,188),(180,187),(181,186),(182,185),(183,184),(208,212),(209,211)]])
112 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 53A | ··· | 53Z | 106A | ··· | 106Z | 212A | ··· | 212AZ |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 53 | ··· | 53 | 106 | ··· | 106 | 212 | ··· | 212 |
| size | 1 | 1 | 53 | 53 | 1 | 1 | 53 | 53 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
112 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C4 | D53 | D106 | C4×D53 |
| kernel | C4×D53 | Dic53 | C212 | D106 | D53 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 26 | 26 | 52 |
Matrix representation of C4×D53 ►in GL3(𝔽1061) generated by
| 103 | 0 | 0 |
| 0 | 1060 | 0 |
| 0 | 0 | 1060 |
| 1 | 0 | 0 |
| 0 | 75 | 1 |
| 0 | 295 | 202 |
| 1060 | 0 | 0 |
| 0 | 261 | 968 |
| 0 | 105 | 800 |
G:=sub<GL(3,GF(1061))| [103,0,0,0,1060,0,0,0,1060],[1,0,0,0,75,295,0,1,202],[1060,0,0,0,261,105,0,968,800] >;
C4×D53 in GAP, Magma, Sage, TeX
C_4\times D_{53} % in TeX
G:=Group("C4xD53"); // GroupNames label
G:=SmallGroup(424,5);
// by ID
G=gap.SmallGroup(424,5);
# by ID
G:=PCGroup([4,-2,-2,-2,-53,21,6659]);
// Polycyclic
G:=Group<a,b,c|a^4=b^53=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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