direct product, abelian, monomial, 2-elementary
Aliases: C4×C52, SmallGroup(208,20)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C4×C52 |
| C1 — C4×C52 |
| C1 — C4×C52 |
Generators and relations for C4×C52
G = < a,b | a4=b52=1, ab=ba >
Smallest permutation representation of C4×C52
►Regular action on 208 points
Generators in S208
(1 65 202 118)(2 66 203 119)(3 67 204 120)(4 68 205 121)(5 69 206 122)(6 70 207 123)(7 71 208 124)(8 72 157 125)(9 73 158 126)(10 74 159 127)(11 75 160 128)(12 76 161 129)(13 77 162 130)(14 78 163 131)(15 79 164 132)(16 80 165 133)(17 81 166 134)(18 82 167 135)(19 83 168 136)(20 84 169 137)(21 85 170 138)(22 86 171 139)(23 87 172 140)(24 88 173 141)(25 89 174 142)(26 90 175 143)(27 91 176 144)(28 92 177 145)(29 93 178 146)(30 94 179 147)(31 95 180 148)(32 96 181 149)(33 97 182 150)(34 98 183 151)(35 99 184 152)(36 100 185 153)(37 101 186 154)(38 102 187 155)(39 103 188 156)(40 104 189 105)(41 53 190 106)(42 54 191 107)(43 55 192 108)(44 56 193 109)(45 57 194 110)(46 58 195 111)(47 59 196 112)(48 60 197 113)(49 61 198 114)(50 62 199 115)(51 63 200 116)(52 64 201 117)
(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)
G:=sub<Sym(208)| (1,65,202,118)(2,66,203,119)(3,67,204,120)(4,68,205,121)(5,69,206,122)(6,70,207,123)(7,71,208,124)(8,72,157,125)(9,73,158,126)(10,74,159,127)(11,75,160,128)(12,76,161,129)(13,77,162,130)(14,78,163,131)(15,79,164,132)(16,80,165,133)(17,81,166,134)(18,82,167,135)(19,83,168,136)(20,84,169,137)(21,85,170,138)(22,86,171,139)(23,87,172,140)(24,88,173,141)(25,89,174,142)(26,90,175,143)(27,91,176,144)(28,92,177,145)(29,93,178,146)(30,94,179,147)(31,95,180,148)(32,96,181,149)(33,97,182,150)(34,98,183,151)(35,99,184,152)(36,100,185,153)(37,101,186,154)(38,102,187,155)(39,103,188,156)(40,104,189,105)(41,53,190,106)(42,54,191,107)(43,55,192,108)(44,56,193,109)(45,57,194,110)(46,58,195,111)(47,59,196,112)(48,60,197,113)(49,61,198,114)(50,62,199,115)(51,63,200,116)(52,64,201,117), (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)>;
G:=Group( (1,65,202,118)(2,66,203,119)(3,67,204,120)(4,68,205,121)(5,69,206,122)(6,70,207,123)(7,71,208,124)(8,72,157,125)(9,73,158,126)(10,74,159,127)(11,75,160,128)(12,76,161,129)(13,77,162,130)(14,78,163,131)(15,79,164,132)(16,80,165,133)(17,81,166,134)(18,82,167,135)(19,83,168,136)(20,84,169,137)(21,85,170,138)(22,86,171,139)(23,87,172,140)(24,88,173,141)(25,89,174,142)(26,90,175,143)(27,91,176,144)(28,92,177,145)(29,93,178,146)(30,94,179,147)(31,95,180,148)(32,96,181,149)(33,97,182,150)(34,98,183,151)(35,99,184,152)(36,100,185,153)(37,101,186,154)(38,102,187,155)(39,103,188,156)(40,104,189,105)(41,53,190,106)(42,54,191,107)(43,55,192,108)(44,56,193,109)(45,57,194,110)(46,58,195,111)(47,59,196,112)(48,60,197,113)(49,61,198,114)(50,62,199,115)(51,63,200,116)(52,64,201,117), (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) );
G=PermutationGroup([[(1,65,202,118),(2,66,203,119),(3,67,204,120),(4,68,205,121),(5,69,206,122),(6,70,207,123),(7,71,208,124),(8,72,157,125),(9,73,158,126),(10,74,159,127),(11,75,160,128),(12,76,161,129),(13,77,162,130),(14,78,163,131),(15,79,164,132),(16,80,165,133),(17,81,166,134),(18,82,167,135),(19,83,168,136),(20,84,169,137),(21,85,170,138),(22,86,171,139),(23,87,172,140),(24,88,173,141),(25,89,174,142),(26,90,175,143),(27,91,176,144),(28,92,177,145),(29,93,178,146),(30,94,179,147),(31,95,180,148),(32,96,181,149),(33,97,182,150),(34,98,183,151),(35,99,184,152),(36,100,185,153),(37,101,186,154),(38,102,187,155),(39,103,188,156),(40,104,189,105),(41,53,190,106),(42,54,191,107),(43,55,192,108),(44,56,193,109),(45,57,194,110),(46,58,195,111),(47,59,196,112),(48,60,197,113),(49,61,198,114),(50,62,199,115),(51,63,200,116),(52,64,201,117)], [(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)]])
C4×C52 is a maximal subgroup of
C26.7C42 C52⋊3C8 D52⋊4C4 C52⋊2Q8 C52.6Q8 C42⋊D13 C4⋊D52 C4.D52 C42⋊2D13
208 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4L | 13A | ··· | 13L | 26A | ··· | 26AJ | 52A | ··· | 52EN |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
208 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||
| image | C1 | C2 | C4 | C13 | C26 | C52 |
| kernel | C4×C52 | C2×C52 | C52 | C42 | C2×C4 | C4 |
| # reps | 1 | 3 | 12 | 12 | 36 | 144 |
Matrix representation of C4×C52 ►in GL2(𝔽53) generated by
| 23 | 0 |
| 0 | 23 |
| 19 | 0 |
| 0 | 28 |
G:=sub<GL(2,GF(53))| [23,0,0,23],[19,0,0,28] >;
C4×C52 in GAP, Magma, Sage, TeX
C_4\times C_{52} % in TeX
G:=Group("C4xC52"); // GroupNames label
G:=SmallGroup(208,20);
// by ID
G=gap.SmallGroup(208,20);
# by ID
G:=PCGroup([5,-2,-2,-13,-2,-2,260,526]);
// Polycyclic
G:=Group<a,b|a^4=b^52=1,a*b=b*a>;
// generators/relations
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