metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C52.1C8, C13⋊2M5(2), C4.(C13⋊C8), C13⋊C16⋊2C2, (C2×C52).7C4, (C2×C26).2C8, C26.6(C2×C8), C22.(C13⋊C8), C13⋊2C8.6C4, C52.18(C2×C4), C13⋊2C8.17C22, C2.3(C2×C13⋊C8), C4.19(C2×C13⋊C4), (C2×C4).5(C13⋊C4), (C2×C13⋊2C8).11C2, SmallGroup(416,73)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C52.C8
G = < a,b | a52=1, b8=a26, bab-1=a31 >
Smallest permutation representation of C52.C8
►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)
(1 195 83 124 14 182 96 111 27 169 57 150 40 208 70 137)(2 190 56 155 15 177 69 142 28 164 82 129 41 203 95 116)(3 185 81 134 16 172 94 121 29 159 55 108 42 198 68 147)(4 180 54 113 17 167 67 152 30 206 80 139 43 193 93 126)(5 175 79 144 18 162 92 131 31 201 53 118 44 188 66 105)(6 170 104 123 19 157 65 110 32 196 78 149 45 183 91 136)(7 165 77 154 20 204 90 141 33 191 103 128 46 178 64 115)(8 160 102 133 21 199 63 120 34 186 76 107 47 173 89 146)(9 207 75 112 22 194 88 151 35 181 101 138 48 168 62 125)(10 202 100 143 23 189 61 130 36 176 74 117 49 163 87 156)(11 197 73 122 24 184 86 109 37 171 99 148 50 158 60 135)(12 192 98 153 25 179 59 140 38 166 72 127 51 205 85 114)(13 187 71 132 26 174 84 119 39 161 97 106 52 200 58 145)
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), (1,195,83,124,14,182,96,111,27,169,57,150,40,208,70,137)(2,190,56,155,15,177,69,142,28,164,82,129,41,203,95,116)(3,185,81,134,16,172,94,121,29,159,55,108,42,198,68,147)(4,180,54,113,17,167,67,152,30,206,80,139,43,193,93,126)(5,175,79,144,18,162,92,131,31,201,53,118,44,188,66,105)(6,170,104,123,19,157,65,110,32,196,78,149,45,183,91,136)(7,165,77,154,20,204,90,141,33,191,103,128,46,178,64,115)(8,160,102,133,21,199,63,120,34,186,76,107,47,173,89,146)(9,207,75,112,22,194,88,151,35,181,101,138,48,168,62,125)(10,202,100,143,23,189,61,130,36,176,74,117,49,163,87,156)(11,197,73,122,24,184,86,109,37,171,99,148,50,158,60,135)(12,192,98,153,25,179,59,140,38,166,72,127,51,205,85,114)(13,187,71,132,26,174,84,119,39,161,97,106,52,200,58,145)>;
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), (1,195,83,124,14,182,96,111,27,169,57,150,40,208,70,137)(2,190,56,155,15,177,69,142,28,164,82,129,41,203,95,116)(3,185,81,134,16,172,94,121,29,159,55,108,42,198,68,147)(4,180,54,113,17,167,67,152,30,206,80,139,43,193,93,126)(5,175,79,144,18,162,92,131,31,201,53,118,44,188,66,105)(6,170,104,123,19,157,65,110,32,196,78,149,45,183,91,136)(7,165,77,154,20,204,90,141,33,191,103,128,46,178,64,115)(8,160,102,133,21,199,63,120,34,186,76,107,47,173,89,146)(9,207,75,112,22,194,88,151,35,181,101,138,48,168,62,125)(10,202,100,143,23,189,61,130,36,176,74,117,49,163,87,156)(11,197,73,122,24,184,86,109,37,171,99,148,50,158,60,135)(12,192,98,153,25,179,59,140,38,166,72,127,51,205,85,114)(13,187,71,132,26,174,84,119,39,161,97,106,52,200,58,145) );
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)], [(1,195,83,124,14,182,96,111,27,169,57,150,40,208,70,137),(2,190,56,155,15,177,69,142,28,164,82,129,41,203,95,116),(3,185,81,134,16,172,94,121,29,159,55,108,42,198,68,147),(4,180,54,113,17,167,67,152,30,206,80,139,43,193,93,126),(5,175,79,144,18,162,92,131,31,201,53,118,44,188,66,105),(6,170,104,123,19,157,65,110,32,196,78,149,45,183,91,136),(7,165,77,154,20,204,90,141,33,191,103,128,46,178,64,115),(8,160,102,133,21,199,63,120,34,186,76,107,47,173,89,146),(9,207,75,112,22,194,88,151,35,181,101,138,48,168,62,125),(10,202,100,143,23,189,61,130,36,176,74,117,49,163,87,156),(11,197,73,122,24,184,86,109,37,171,99,148,50,158,60,135),(12,192,98,153,25,179,59,140,38,166,72,127,51,205,85,114),(13,187,71,132,26,174,84,119,39,161,97,106,52,200,58,145)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 8E | 8F | 13A | 13B | 13C | 16A | ··· | 16H | 26A | ··· | 26I | 52A | ··· | 52L |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 13 | 13 | 13 | 16 | ··· | 16 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 13 | 13 | 13 | 13 | 26 | 26 | 4 | 4 | 4 | 26 | ··· | 26 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | - | ||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | M5(2) | C13⋊C4 | C13⋊C8 | C2×C13⋊C4 | C13⋊C8 | C52.C8 |
| kernel | C52.C8 | C13⋊C16 | C2×C13⋊2C8 | C13⋊2C8 | C2×C52 | C52 | C2×C26 | C13 | C2×C4 | C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 3 | 3 | 3 | 3 | 12 |
Matrix representation of C52.C8 ►in GL6(𝔽1249)
| 585 | 1014 | 0 | 0 | 0 | 0 |
| 0 | 664 | 0 | 0 | 0 | 0 |
| 0 | 0 | 75 | 610 | 1204 | 564 |
| 0 | 0 | 640 | 1 | 0 | 1248 |
| 0 | 0 | 1 | 640 | 566 | 1247 |
| 0 | 0 | 2 | 640 | 566 | 1247 |
| 861 | 254 | 0 | 0 | 0 | 0 |
| 832 | 388 | 0 | 0 | 0 | 0 |
| 0 | 0 | 909 | 798 | 346 | 907 |
| 0 | 0 | 583 | 562 | 447 | 865 |
| 0 | 0 | 853 | 107 | 525 | 645 |
| 0 | 0 | 240 | 1141 | 1127 | 502 |
G:=sub<GL(6,GF(1249))| [585,0,0,0,0,0,1014,664,0,0,0,0,0,0,75,640,1,2,0,0,610,1,640,640,0,0,1204,0,566,566,0,0,564,1248,1247,1247],[861,832,0,0,0,0,254,388,0,0,0,0,0,0,909,583,853,240,0,0,798,562,107,1141,0,0,346,447,525,1127,0,0,907,865,645,502] >;
C52.C8 in GAP, Magma, Sage, TeX
C_{52}.C_8 % in TeX
G:=Group("C52.C8"); // GroupNames label
G:=SmallGroup(416,73);
// by ID
G=gap.SmallGroup(416,73);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,217,50,69,9221,3473]);
// Polycyclic
G:=Group<a,b|a^52=1,b^8=a^26,b*a*b^-1=a^31>;
// generators/relations
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