metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C52.55D4, C26.11M4(2), C23.2Dic13, (C2×C26)⋊4C8, C13⋊4(C22⋊C8), C22⋊(C13⋊2C8), (C2×C52).15C4, C26.19(C2×C8), (C2×C4).94D26, (C22×C26).9C4, (C2×C4).4Dic13, (C22×C4).1D13, C4.30(C13⋊D4), (C22×C52).11C2, C2.3(C52.4C4), C26.23(C22⋊C4), (C2×C52).108C22, C2.1(C23.D13), C22.9(C2×Dic13), C2.5(C2×C13⋊2C8), (C2×C13⋊2C8)⋊10C2, (C2×C26).47(C2×C4), SmallGroup(416,37)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C52.55D4
G = < a,b,c | a52=1, b4=a26, c2=a13, bab-1=cac-1=a25, cbc-1=a39b3 >
Smallest permutation representation of C52.55D4
►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)
(1 181 67 155 27 207 93 129)(2 206 68 128 28 180 94 154)(3 179 69 153 29 205 95 127)(4 204 70 126 30 178 96 152)(5 177 71 151 31 203 97 125)(6 202 72 124 32 176 98 150)(7 175 73 149 33 201 99 123)(8 200 74 122 34 174 100 148)(9 173 75 147 35 199 101 121)(10 198 76 120 36 172 102 146)(11 171 77 145 37 197 103 119)(12 196 78 118 38 170 104 144)(13 169 79 143 39 195 53 117)(14 194 80 116 40 168 54 142)(15 167 81 141 41 193 55 115)(16 192 82 114 42 166 56 140)(17 165 83 139 43 191 57 113)(18 190 84 112 44 164 58 138)(19 163 85 137 45 189 59 111)(20 188 86 110 46 162 60 136)(21 161 87 135 47 187 61 109)(22 186 88 108 48 160 62 134)(23 159 89 133 49 185 63 107)(24 184 90 106 50 158 64 132)(25 157 91 131 51 183 65 105)(26 182 92 156 52 208 66 130)
(1 142 14 155 27 116 40 129)(2 115 15 128 28 141 41 154)(3 140 16 153 29 114 42 127)(4 113 17 126 30 139 43 152)(5 138 18 151 31 112 44 125)(6 111 19 124 32 137 45 150)(7 136 20 149 33 110 46 123)(8 109 21 122 34 135 47 148)(9 134 22 147 35 108 48 121)(10 107 23 120 36 133 49 146)(11 132 24 145 37 106 50 119)(12 105 25 118 38 131 51 144)(13 130 26 143 39 156 52 117)(53 208 66 169 79 182 92 195)(54 181 67 194 80 207 93 168)(55 206 68 167 81 180 94 193)(56 179 69 192 82 205 95 166)(57 204 70 165 83 178 96 191)(58 177 71 190 84 203 97 164)(59 202 72 163 85 176 98 189)(60 175 73 188 86 201 99 162)(61 200 74 161 87 174 100 187)(62 173 75 186 88 199 101 160)(63 198 76 159 89 172 102 185)(64 171 77 184 90 197 103 158)(65 196 78 157 91 170 104 183)
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), (1,181,67,155,27,207,93,129)(2,206,68,128,28,180,94,154)(3,179,69,153,29,205,95,127)(4,204,70,126,30,178,96,152)(5,177,71,151,31,203,97,125)(6,202,72,124,32,176,98,150)(7,175,73,149,33,201,99,123)(8,200,74,122,34,174,100,148)(9,173,75,147,35,199,101,121)(10,198,76,120,36,172,102,146)(11,171,77,145,37,197,103,119)(12,196,78,118,38,170,104,144)(13,169,79,143,39,195,53,117)(14,194,80,116,40,168,54,142)(15,167,81,141,41,193,55,115)(16,192,82,114,42,166,56,140)(17,165,83,139,43,191,57,113)(18,190,84,112,44,164,58,138)(19,163,85,137,45,189,59,111)(20,188,86,110,46,162,60,136)(21,161,87,135,47,187,61,109)(22,186,88,108,48,160,62,134)(23,159,89,133,49,185,63,107)(24,184,90,106,50,158,64,132)(25,157,91,131,51,183,65,105)(26,182,92,156,52,208,66,130), (1,142,14,155,27,116,40,129)(2,115,15,128,28,141,41,154)(3,140,16,153,29,114,42,127)(4,113,17,126,30,139,43,152)(5,138,18,151,31,112,44,125)(6,111,19,124,32,137,45,150)(7,136,20,149,33,110,46,123)(8,109,21,122,34,135,47,148)(9,134,22,147,35,108,48,121)(10,107,23,120,36,133,49,146)(11,132,24,145,37,106,50,119)(12,105,25,118,38,131,51,144)(13,130,26,143,39,156,52,117)(53,208,66,169,79,182,92,195)(54,181,67,194,80,207,93,168)(55,206,68,167,81,180,94,193)(56,179,69,192,82,205,95,166)(57,204,70,165,83,178,96,191)(58,177,71,190,84,203,97,164)(59,202,72,163,85,176,98,189)(60,175,73,188,86,201,99,162)(61,200,74,161,87,174,100,187)(62,173,75,186,88,199,101,160)(63,198,76,159,89,172,102,185)(64,171,77,184,90,197,103,158)(65,196,78,157,91,170,104,183)>;
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), (1,181,67,155,27,207,93,129)(2,206,68,128,28,180,94,154)(3,179,69,153,29,205,95,127)(4,204,70,126,30,178,96,152)(5,177,71,151,31,203,97,125)(6,202,72,124,32,176,98,150)(7,175,73,149,33,201,99,123)(8,200,74,122,34,174,100,148)(9,173,75,147,35,199,101,121)(10,198,76,120,36,172,102,146)(11,171,77,145,37,197,103,119)(12,196,78,118,38,170,104,144)(13,169,79,143,39,195,53,117)(14,194,80,116,40,168,54,142)(15,167,81,141,41,193,55,115)(16,192,82,114,42,166,56,140)(17,165,83,139,43,191,57,113)(18,190,84,112,44,164,58,138)(19,163,85,137,45,189,59,111)(20,188,86,110,46,162,60,136)(21,161,87,135,47,187,61,109)(22,186,88,108,48,160,62,134)(23,159,89,133,49,185,63,107)(24,184,90,106,50,158,64,132)(25,157,91,131,51,183,65,105)(26,182,92,156,52,208,66,130), (1,142,14,155,27,116,40,129)(2,115,15,128,28,141,41,154)(3,140,16,153,29,114,42,127)(4,113,17,126,30,139,43,152)(5,138,18,151,31,112,44,125)(6,111,19,124,32,137,45,150)(7,136,20,149,33,110,46,123)(8,109,21,122,34,135,47,148)(9,134,22,147,35,108,48,121)(10,107,23,120,36,133,49,146)(11,132,24,145,37,106,50,119)(12,105,25,118,38,131,51,144)(13,130,26,143,39,156,52,117)(53,208,66,169,79,182,92,195)(54,181,67,194,80,207,93,168)(55,206,68,167,81,180,94,193)(56,179,69,192,82,205,95,166)(57,204,70,165,83,178,96,191)(58,177,71,190,84,203,97,164)(59,202,72,163,85,176,98,189)(60,175,73,188,86,201,99,162)(61,200,74,161,87,174,100,187)(62,173,75,186,88,199,101,160)(63,198,76,159,89,172,102,185)(64,171,77,184,90,197,103,158)(65,196,78,157,91,170,104,183) );
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)], [(1,181,67,155,27,207,93,129),(2,206,68,128,28,180,94,154),(3,179,69,153,29,205,95,127),(4,204,70,126,30,178,96,152),(5,177,71,151,31,203,97,125),(6,202,72,124,32,176,98,150),(7,175,73,149,33,201,99,123),(8,200,74,122,34,174,100,148),(9,173,75,147,35,199,101,121),(10,198,76,120,36,172,102,146),(11,171,77,145,37,197,103,119),(12,196,78,118,38,170,104,144),(13,169,79,143,39,195,53,117),(14,194,80,116,40,168,54,142),(15,167,81,141,41,193,55,115),(16,192,82,114,42,166,56,140),(17,165,83,139,43,191,57,113),(18,190,84,112,44,164,58,138),(19,163,85,137,45,189,59,111),(20,188,86,110,46,162,60,136),(21,161,87,135,47,187,61,109),(22,186,88,108,48,160,62,134),(23,159,89,133,49,185,63,107),(24,184,90,106,50,158,64,132),(25,157,91,131,51,183,65,105),(26,182,92,156,52,208,66,130)], [(1,142,14,155,27,116,40,129),(2,115,15,128,28,141,41,154),(3,140,16,153,29,114,42,127),(4,113,17,126,30,139,43,152),(5,138,18,151,31,112,44,125),(6,111,19,124,32,137,45,150),(7,136,20,149,33,110,46,123),(8,109,21,122,34,135,47,148),(9,134,22,147,35,108,48,121),(10,107,23,120,36,133,49,146),(11,132,24,145,37,106,50,119),(12,105,25,118,38,131,51,144),(13,130,26,143,39,156,52,117),(53,208,66,169,79,182,92,195),(54,181,67,194,80,207,93,168),(55,206,68,167,81,180,94,193),(56,179,69,192,82,205,95,166),(57,204,70,165,83,178,96,191),(58,177,71,190,84,203,97,164),(59,202,72,163,85,176,98,189),(60,175,73,188,86,201,99,162),(61,200,74,161,87,174,100,187),(62,173,75,186,88,199,101,160),(63,198,76,159,89,172,102,185),(64,171,77,184,90,197,103,158),(65,196,78,157,91,170,104,183)]])
116 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 8A | ··· | 8H | 13A | ··· | 13F | 26A | ··· | 26AP | 52A | ··· | 52AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 26 | ··· | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
116 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | |||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | D4 | M4(2) | D13 | Dic13 | D26 | Dic13 | C13⋊D4 | C13⋊2C8 | C52.4C4 |
| kernel | C52.55D4 | C2×C13⋊2C8 | C22×C52 | C2×C52 | C22×C26 | C2×C26 | C52 | C26 | C22×C4 | C2×C4 | C2×C4 | C23 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 2 | 2 | 6 | 6 | 6 | 6 | 24 | 24 | 24 |
Matrix representation of C52.55D4 ►in GL3(𝔽313) generated by
| 288 | 0 | 0 |
| 0 | 278 | 0 |
| 0 | 173 | 161 |
| 188 | 0 | 0 |
| 0 | 81 | 137 |
| 0 | 105 | 232 |
| 125 | 0 | 0 |
| 0 | 81 | 137 |
| 0 | 279 | 232 |
G:=sub<GL(3,GF(313))| [288,0,0,0,278,173,0,0,161],[188,0,0,0,81,105,0,137,232],[125,0,0,0,81,279,0,137,232] >;
C52.55D4 in GAP, Magma, Sage, TeX
C_{52}._{55}D_4 % in TeX
G:=Group("C52.55D4"); // GroupNames label
G:=SmallGroup(416,37);
// by ID
G=gap.SmallGroup(416,37);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,86,13829]);
// 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^39*b^3>;
// generators/relations
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