metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C52⋊3C4, C4⋊Dic13, C26.4D4, C2.1D52, C26.2Q8, C2.2Dic26, C22.5D26, C13⋊3(C4⋊C4), (C2×C52).3C2, (C2×C4).3D13, C26.15(C2×C4), (C2×C26).5C22, C2.4(C2×Dic13), (C2×Dic13).2C2, SmallGroup(208,13)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C52⋊3C4
G = < a,b | a52=b4=1, bab-1=a-1 >
Smallest permutation representation of C52⋊3C4
►Regular action on 208 points
Generators in S208
(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)
(1 183 91 120)(2 182 92 119)(3 181 93 118)(4 180 94 117)(5 179 95 116)(6 178 96 115)(7 177 97 114)(8 176 98 113)(9 175 99 112)(10 174 100 111)(11 173 101 110)(12 172 102 109)(13 171 103 108)(14 170 104 107)(15 169 53 106)(16 168 54 105)(17 167 55 156)(18 166 56 155)(19 165 57 154)(20 164 58 153)(21 163 59 152)(22 162 60 151)(23 161 61 150)(24 160 62 149)(25 159 63 148)(26 158 64 147)(27 157 65 146)(28 208 66 145)(29 207 67 144)(30 206 68 143)(31 205 69 142)(32 204 70 141)(33 203 71 140)(34 202 72 139)(35 201 73 138)(36 200 74 137)(37 199 75 136)(38 198 76 135)(39 197 77 134)(40 196 78 133)(41 195 79 132)(42 194 80 131)(43 193 81 130)(44 192 82 129)(45 191 83 128)(46 190 84 127)(47 189 85 126)(48 188 86 125)(49 187 87 124)(50 186 88 123)(51 185 89 122)(52 184 90 121)
G:=sub<Sym(208)| (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), (1,183,91,120)(2,182,92,119)(3,181,93,118)(4,180,94,117)(5,179,95,116)(6,178,96,115)(7,177,97,114)(8,176,98,113)(9,175,99,112)(10,174,100,111)(11,173,101,110)(12,172,102,109)(13,171,103,108)(14,170,104,107)(15,169,53,106)(16,168,54,105)(17,167,55,156)(18,166,56,155)(19,165,57,154)(20,164,58,153)(21,163,59,152)(22,162,60,151)(23,161,61,150)(24,160,62,149)(25,159,63,148)(26,158,64,147)(27,157,65,146)(28,208,66,145)(29,207,67,144)(30,206,68,143)(31,205,69,142)(32,204,70,141)(33,203,71,140)(34,202,72,139)(35,201,73,138)(36,200,74,137)(37,199,75,136)(38,198,76,135)(39,197,77,134)(40,196,78,133)(41,195,79,132)(42,194,80,131)(43,193,81,130)(44,192,82,129)(45,191,83,128)(46,190,84,127)(47,189,85,126)(48,188,86,125)(49,187,87,124)(50,186,88,123)(51,185,89,122)(52,184,90,121)>;
G:=Group( (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), (1,183,91,120)(2,182,92,119)(3,181,93,118)(4,180,94,117)(5,179,95,116)(6,178,96,115)(7,177,97,114)(8,176,98,113)(9,175,99,112)(10,174,100,111)(11,173,101,110)(12,172,102,109)(13,171,103,108)(14,170,104,107)(15,169,53,106)(16,168,54,105)(17,167,55,156)(18,166,56,155)(19,165,57,154)(20,164,58,153)(21,163,59,152)(22,162,60,151)(23,161,61,150)(24,160,62,149)(25,159,63,148)(26,158,64,147)(27,157,65,146)(28,208,66,145)(29,207,67,144)(30,206,68,143)(31,205,69,142)(32,204,70,141)(33,203,71,140)(34,202,72,139)(35,201,73,138)(36,200,74,137)(37,199,75,136)(38,198,76,135)(39,197,77,134)(40,196,78,133)(41,195,79,132)(42,194,80,131)(43,193,81,130)(44,192,82,129)(45,191,83,128)(46,190,84,127)(47,189,85,126)(48,188,86,125)(49,187,87,124)(50,186,88,123)(51,185,89,122)(52,184,90,121) );
G=PermutationGroup([[(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)], [(1,183,91,120),(2,182,92,119),(3,181,93,118),(4,180,94,117),(5,179,95,116),(6,178,96,115),(7,177,97,114),(8,176,98,113),(9,175,99,112),(10,174,100,111),(11,173,101,110),(12,172,102,109),(13,171,103,108),(14,170,104,107),(15,169,53,106),(16,168,54,105),(17,167,55,156),(18,166,56,155),(19,165,57,154),(20,164,58,153),(21,163,59,152),(22,162,60,151),(23,161,61,150),(24,160,62,149),(25,159,63,148),(26,158,64,147),(27,157,65,146),(28,208,66,145),(29,207,67,144),(30,206,68,143),(31,205,69,142),(32,204,70,141),(33,203,71,140),(34,202,72,139),(35,201,73,138),(36,200,74,137),(37,199,75,136),(38,198,76,135),(39,197,77,134),(40,196,78,133),(41,195,79,132),(42,194,80,131),(43,193,81,130),(44,192,82,129),(45,191,83,128),(46,190,84,127),(47,189,85,126),(48,188,86,125),(49,187,87,124),(50,186,88,123),(51,185,89,122),(52,184,90,121)]])
C52⋊3C4 is a maximal subgroup of
C26.D8 C52.Q8 C52.44D4 C104⋊6C4 C104⋊5C4 D52⋊5C4 D4⋊Dic13 Q8⋊Dic13 C4×Dic26 C52⋊2Q8 C52.6Q8 C4×D52 C22⋊Dic26 C23.D26 D26.12D4 C22.D52 C52⋊Q8 Dic13.Q8 C4.Dic26 C4⋊C4×D13 C4⋊C4⋊7D13 D26⋊2Q8 C4⋊C4⋊D13 C52.48D4 C23.21D26 C52⋊7D4 D4×Dic13 C52⋊2D4 Q8×Dic13 D26⋊3Q8
C52⋊3C4 is a maximal quotient of
C52⋊3C8 C104⋊6C4 C104⋊5C4 C104.6C4 C26.10C42
58 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 13A | ··· | 13F | 26A | ··· | 26R | 52A | ··· | 52X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 26 | 26 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | - | + | |
| image | C1 | C2 | C2 | C4 | D4 | Q8 | D13 | Dic13 | D26 | Dic26 | D52 |
| kernel | C52⋊3C4 | C2×Dic13 | C2×C52 | C52 | C26 | C26 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 6 | 12 | 6 | 12 | 12 |
Matrix representation of C52⋊3C4 ►in GL3(𝔽53) generated by
| 1 | 0 | 0 |
| 0 | 50 | 38 |
| 0 | 31 | 31 |
| 23 | 0 | 0 |
| 0 | 10 | 35 |
| 0 | 32 | 43 |
G:=sub<GL(3,GF(53))| [1,0,0,0,50,31,0,38,31],[23,0,0,0,10,32,0,35,43] >;
C52⋊3C4 in GAP, Magma, Sage, TeX
C_{52}\rtimes_3C_4 % in TeX
G:=Group("C52:3C4"); // GroupNames label
G:=SmallGroup(208,13);
// by ID
G=gap.SmallGroup(208,13);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-13,20,101,46,4804]);
// Polycyclic
G:=Group<a,b|a^52=b^4=1,b*a*b^-1=a^-1>;
// generators/relations
Export