metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C52.53D4, M4(2).1D13, C22.1Dic26, (C2×C26).Q8, C13⋊2C8.1C4, C52.26(C2×C4), (C2×C4).37D26, C4.13(C4×D13), C26.15(C4⋊C4), C13⋊4(C8.C4), C52.4C4.2C2, C4.28(C13⋊D4), (C2×C52).12C22, C2.5(C26.D4), (C13×M4(2)).1C2, (C2×C13⋊2C8).4C2, SmallGroup(416,29)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C52.53D4
G = < a,b,c | a52=1, b4=a26, c2=a13, bab-1=cac-1=a25, cbc-1=a26b3 >
Smallest permutation representation of C52.53D4
►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)
(1 114 40 153 27 140 14 127)(2 139 41 126 28 113 15 152)(3 112 42 151 29 138 16 125)(4 137 43 124 30 111 17 150)(5 110 44 149 31 136 18 123)(6 135 45 122 32 109 19 148)(7 108 46 147 33 134 20 121)(8 133 47 120 34 107 21 146)(9 106 48 145 35 132 22 119)(10 131 49 118 36 105 23 144)(11 156 50 143 37 130 24 117)(12 129 51 116 38 155 25 142)(13 154 52 141 39 128 26 115)(53 158 66 171 79 184 92 197)(54 183 67 196 80 157 93 170)(55 208 68 169 81 182 94 195)(56 181 69 194 82 207 95 168)(57 206 70 167 83 180 96 193)(58 179 71 192 84 205 97 166)(59 204 72 165 85 178 98 191)(60 177 73 190 86 203 99 164)(61 202 74 163 87 176 100 189)(62 175 75 188 88 201 101 162)(63 200 76 161 89 174 102 187)(64 173 77 186 90 199 103 160)(65 198 78 159 91 172 104 185)
(1 158 14 171 27 184 40 197)(2 183 15 196 28 157 41 170)(3 208 16 169 29 182 42 195)(4 181 17 194 30 207 43 168)(5 206 18 167 31 180 44 193)(6 179 19 192 32 205 45 166)(7 204 20 165 33 178 46 191)(8 177 21 190 34 203 47 164)(9 202 22 163 35 176 48 189)(10 175 23 188 36 201 49 162)(11 200 24 161 37 174 50 187)(12 173 25 186 38 199 51 160)(13 198 26 159 39 172 52 185)(53 127 66 140 79 153 92 114)(54 152 67 113 80 126 93 139)(55 125 68 138 81 151 94 112)(56 150 69 111 82 124 95 137)(57 123 70 136 83 149 96 110)(58 148 71 109 84 122 97 135)(59 121 72 134 85 147 98 108)(60 146 73 107 86 120 99 133)(61 119 74 132 87 145 100 106)(62 144 75 105 88 118 101 131)(63 117 76 130 89 143 102 156)(64 142 77 155 90 116 103 129)(65 115 78 128 91 141 104 154)
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), (1,114,40,153,27,140,14,127)(2,139,41,126,28,113,15,152)(3,112,42,151,29,138,16,125)(4,137,43,124,30,111,17,150)(5,110,44,149,31,136,18,123)(6,135,45,122,32,109,19,148)(7,108,46,147,33,134,20,121)(8,133,47,120,34,107,21,146)(9,106,48,145,35,132,22,119)(10,131,49,118,36,105,23,144)(11,156,50,143,37,130,24,117)(12,129,51,116,38,155,25,142)(13,154,52,141,39,128,26,115)(53,158,66,171,79,184,92,197)(54,183,67,196,80,157,93,170)(55,208,68,169,81,182,94,195)(56,181,69,194,82,207,95,168)(57,206,70,167,83,180,96,193)(58,179,71,192,84,205,97,166)(59,204,72,165,85,178,98,191)(60,177,73,190,86,203,99,164)(61,202,74,163,87,176,100,189)(62,175,75,188,88,201,101,162)(63,200,76,161,89,174,102,187)(64,173,77,186,90,199,103,160)(65,198,78,159,91,172,104,185), (1,158,14,171,27,184,40,197)(2,183,15,196,28,157,41,170)(3,208,16,169,29,182,42,195)(4,181,17,194,30,207,43,168)(5,206,18,167,31,180,44,193)(6,179,19,192,32,205,45,166)(7,204,20,165,33,178,46,191)(8,177,21,190,34,203,47,164)(9,202,22,163,35,176,48,189)(10,175,23,188,36,201,49,162)(11,200,24,161,37,174,50,187)(12,173,25,186,38,199,51,160)(13,198,26,159,39,172,52,185)(53,127,66,140,79,153,92,114)(54,152,67,113,80,126,93,139)(55,125,68,138,81,151,94,112)(56,150,69,111,82,124,95,137)(57,123,70,136,83,149,96,110)(58,148,71,109,84,122,97,135)(59,121,72,134,85,147,98,108)(60,146,73,107,86,120,99,133)(61,119,74,132,87,145,100,106)(62,144,75,105,88,118,101,131)(63,117,76,130,89,143,102,156)(64,142,77,155,90,116,103,129)(65,115,78,128,91,141,104,154)>;
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), (1,114,40,153,27,140,14,127)(2,139,41,126,28,113,15,152)(3,112,42,151,29,138,16,125)(4,137,43,124,30,111,17,150)(5,110,44,149,31,136,18,123)(6,135,45,122,32,109,19,148)(7,108,46,147,33,134,20,121)(8,133,47,120,34,107,21,146)(9,106,48,145,35,132,22,119)(10,131,49,118,36,105,23,144)(11,156,50,143,37,130,24,117)(12,129,51,116,38,155,25,142)(13,154,52,141,39,128,26,115)(53,158,66,171,79,184,92,197)(54,183,67,196,80,157,93,170)(55,208,68,169,81,182,94,195)(56,181,69,194,82,207,95,168)(57,206,70,167,83,180,96,193)(58,179,71,192,84,205,97,166)(59,204,72,165,85,178,98,191)(60,177,73,190,86,203,99,164)(61,202,74,163,87,176,100,189)(62,175,75,188,88,201,101,162)(63,200,76,161,89,174,102,187)(64,173,77,186,90,199,103,160)(65,198,78,159,91,172,104,185), (1,158,14,171,27,184,40,197)(2,183,15,196,28,157,41,170)(3,208,16,169,29,182,42,195)(4,181,17,194,30,207,43,168)(5,206,18,167,31,180,44,193)(6,179,19,192,32,205,45,166)(7,204,20,165,33,178,46,191)(8,177,21,190,34,203,47,164)(9,202,22,163,35,176,48,189)(10,175,23,188,36,201,49,162)(11,200,24,161,37,174,50,187)(12,173,25,186,38,199,51,160)(13,198,26,159,39,172,52,185)(53,127,66,140,79,153,92,114)(54,152,67,113,80,126,93,139)(55,125,68,138,81,151,94,112)(56,150,69,111,82,124,95,137)(57,123,70,136,83,149,96,110)(58,148,71,109,84,122,97,135)(59,121,72,134,85,147,98,108)(60,146,73,107,86,120,99,133)(61,119,74,132,87,145,100,106)(62,144,75,105,88,118,101,131)(63,117,76,130,89,143,102,156)(64,142,77,155,90,116,103,129)(65,115,78,128,91,141,104,154) );
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)], [(1,114,40,153,27,140,14,127),(2,139,41,126,28,113,15,152),(3,112,42,151,29,138,16,125),(4,137,43,124,30,111,17,150),(5,110,44,149,31,136,18,123),(6,135,45,122,32,109,19,148),(7,108,46,147,33,134,20,121),(8,133,47,120,34,107,21,146),(9,106,48,145,35,132,22,119),(10,131,49,118,36,105,23,144),(11,156,50,143,37,130,24,117),(12,129,51,116,38,155,25,142),(13,154,52,141,39,128,26,115),(53,158,66,171,79,184,92,197),(54,183,67,196,80,157,93,170),(55,208,68,169,81,182,94,195),(56,181,69,194,82,207,95,168),(57,206,70,167,83,180,96,193),(58,179,71,192,84,205,97,166),(59,204,72,165,85,178,98,191),(60,177,73,190,86,203,99,164),(61,202,74,163,87,176,100,189),(62,175,75,188,88,201,101,162),(63,200,76,161,89,174,102,187),(64,173,77,186,90,199,103,160),(65,198,78,159,91,172,104,185)], [(1,158,14,171,27,184,40,197),(2,183,15,196,28,157,41,170),(3,208,16,169,29,182,42,195),(4,181,17,194,30,207,43,168),(5,206,18,167,31,180,44,193),(6,179,19,192,32,205,45,166),(7,204,20,165,33,178,46,191),(8,177,21,190,34,203,47,164),(9,202,22,163,35,176,48,189),(10,175,23,188,36,201,49,162),(11,200,24,161,37,174,50,187),(12,173,25,186,38,199,51,160),(13,198,26,159,39,172,52,185),(53,127,66,140,79,153,92,114),(54,152,67,113,80,126,93,139),(55,125,68,138,81,151,94,112),(56,150,69,111,82,124,95,137),(57,123,70,136,83,149,96,110),(58,148,71,109,84,122,97,135),(59,121,72,134,85,147,98,108),(60,146,73,107,86,120,99,133),(61,119,74,132,87,145,100,106),(62,144,75,105,88,118,101,131),(63,117,76,130,89,143,102,156),(64,142,77,155,90,116,103,129),(65,115,78,128,91,141,104,154)]])
74 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 13A | ··· | 13F | 26A | ··· | 26F | 26G | ··· | 26L | 52A | ··· | 52L | 52M | ··· | 52R | 104A | ··· | 104X |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 4 | 4 | 26 | 26 | 26 | 26 | 52 | 52 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | - | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | D4 | Q8 | D13 | C8.C4 | D26 | C4×D13 | C13⋊D4 | Dic26 | C52.53D4 |
| kernel | C52.53D4 | C2×C13⋊2C8 | C52.4C4 | C13×M4(2) | C13⋊2C8 | C52 | C2×C26 | M4(2) | C13 | C2×C4 | C4 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 6 | 4 | 6 | 12 | 12 | 12 | 12 |
Matrix representation of C52.53D4 ►in GL4(𝔽313) generated by
| 25 | 0 | 0 | 0 |
| 0 | 25 | 0 | 0 |
| 0 | 0 | 126 | 25 |
| 0 | 0 | 16 | 110 |
| 125 | 0 | 0 | 0 |
| 0 | 308 | 0 | 0 |
| 0 | 0 | 211 | 130 |
| 0 | 0 | 245 | 102 |
| 0 | 308 | 0 | 0 |
| 308 | 0 | 0 | 0 |
| 0 | 0 | 233 | 41 |
| 0 | 0 | 50 | 80 |
G:=sub<GL(4,GF(313))| [25,0,0,0,0,25,0,0,0,0,126,16,0,0,25,110],[125,0,0,0,0,308,0,0,0,0,211,245,0,0,130,102],[0,308,0,0,308,0,0,0,0,0,233,50,0,0,41,80] >;
C52.53D4 in GAP, Magma, Sage, TeX
C_{52}._{53}D_4 % in TeX
G:=Group("C52.53D4"); // GroupNames label
G:=SmallGroup(416,29);
// by ID
G=gap.SmallGroup(416,29);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,121,31,86,297,69,13829]);
// Polycyclic
G:=Group<a,b,c|a^52=1,b^4=a^26,c^2=a^13,b*a*b^-1=c*a*c^-1=a^25,c*b*c^-1=a^26*b^3>;
// generators/relations
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