metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C52.4C8, C104.9C4, C8.22D26, C13⋊4M5(2), C8.2Dic13, C104.22C22, C4.(C13⋊2C8), (C2×C26).5C8, (C2×C8).7D13, C13⋊2C16⋊5C2, C26.18(C2×C8), (C2×C52).19C4, C52.60(C2×C4), (C2×C104).10C2, C22.(C13⋊2C8), (C2×C4).5Dic13, C4.11(C2×Dic13), C2.4(C2×C13⋊2C8), SmallGroup(416,19)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C52.4C8
G = < a,b | a52=1, b8=a26, bab-1=a-1 >
Smallest permutation representation of C52.4C8
►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)
(1 172 58 145 14 159 71 132 27 198 84 119 40 185 97 106)(2 171 59 144 15 158 72 131 28 197 85 118 41 184 98 105)(3 170 60 143 16 157 73 130 29 196 86 117 42 183 99 156)(4 169 61 142 17 208 74 129 30 195 87 116 43 182 100 155)(5 168 62 141 18 207 75 128 31 194 88 115 44 181 101 154)(6 167 63 140 19 206 76 127 32 193 89 114 45 180 102 153)(7 166 64 139 20 205 77 126 33 192 90 113 46 179 103 152)(8 165 65 138 21 204 78 125 34 191 91 112 47 178 104 151)(9 164 66 137 22 203 79 124 35 190 92 111 48 177 53 150)(10 163 67 136 23 202 80 123 36 189 93 110 49 176 54 149)(11 162 68 135 24 201 81 122 37 188 94 109 50 175 55 148)(12 161 69 134 25 200 82 121 38 187 95 108 51 174 56 147)(13 160 70 133 26 199 83 120 39 186 96 107 52 173 57 146)
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), (1,172,58,145,14,159,71,132,27,198,84,119,40,185,97,106)(2,171,59,144,15,158,72,131,28,197,85,118,41,184,98,105)(3,170,60,143,16,157,73,130,29,196,86,117,42,183,99,156)(4,169,61,142,17,208,74,129,30,195,87,116,43,182,100,155)(5,168,62,141,18,207,75,128,31,194,88,115,44,181,101,154)(6,167,63,140,19,206,76,127,32,193,89,114,45,180,102,153)(7,166,64,139,20,205,77,126,33,192,90,113,46,179,103,152)(8,165,65,138,21,204,78,125,34,191,91,112,47,178,104,151)(9,164,66,137,22,203,79,124,35,190,92,111,48,177,53,150)(10,163,67,136,23,202,80,123,36,189,93,110,49,176,54,149)(11,162,68,135,24,201,81,122,37,188,94,109,50,175,55,148)(12,161,69,134,25,200,82,121,38,187,95,108,51,174,56,147)(13,160,70,133,26,199,83,120,39,186,96,107,52,173,57,146)>;
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), (1,172,58,145,14,159,71,132,27,198,84,119,40,185,97,106)(2,171,59,144,15,158,72,131,28,197,85,118,41,184,98,105)(3,170,60,143,16,157,73,130,29,196,86,117,42,183,99,156)(4,169,61,142,17,208,74,129,30,195,87,116,43,182,100,155)(5,168,62,141,18,207,75,128,31,194,88,115,44,181,101,154)(6,167,63,140,19,206,76,127,32,193,89,114,45,180,102,153)(7,166,64,139,20,205,77,126,33,192,90,113,46,179,103,152)(8,165,65,138,21,204,78,125,34,191,91,112,47,178,104,151)(9,164,66,137,22,203,79,124,35,190,92,111,48,177,53,150)(10,163,67,136,23,202,80,123,36,189,93,110,49,176,54,149)(11,162,68,135,24,201,81,122,37,188,94,109,50,175,55,148)(12,161,69,134,25,200,82,121,38,187,95,108,51,174,56,147)(13,160,70,133,26,199,83,120,39,186,96,107,52,173,57,146) );
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)], [(1,172,58,145,14,159,71,132,27,198,84,119,40,185,97,106),(2,171,59,144,15,158,72,131,28,197,85,118,41,184,98,105),(3,170,60,143,16,157,73,130,29,196,86,117,42,183,99,156),(4,169,61,142,17,208,74,129,30,195,87,116,43,182,100,155),(5,168,62,141,18,207,75,128,31,194,88,115,44,181,101,154),(6,167,63,140,19,206,76,127,32,193,89,114,45,180,102,153),(7,166,64,139,20,205,77,126,33,192,90,113,46,179,103,152),(8,165,65,138,21,204,78,125,34,191,91,112,47,178,104,151),(9,164,66,137,22,203,79,124,35,190,92,111,48,177,53,150),(10,163,67,136,23,202,80,123,36,189,93,110,49,176,54,149),(11,162,68,135,24,201,81,122,37,188,94,109,50,175,55,148),(12,161,69,134,25,200,82,121,38,187,95,108,51,174,56,147),(13,160,70,133,26,199,83,120,39,186,96,107,52,173,57,146)]])
116 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 8E | 8F | 13A | ··· | 13F | 16A | ··· | 16H | 26A | ··· | 26R | 52A | ··· | 52X | 104A | ··· | 104AV |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 13 | ··· | 13 | 16 | ··· | 16 | 26 | ··· | 26 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 26 | ··· | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
116 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | ||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | D13 | M5(2) | Dic13 | D26 | Dic13 | C13⋊2C8 | C13⋊2C8 | C52.4C8 |
| kernel | C52.4C8 | C13⋊2C16 | C2×C104 | C104 | C2×C52 | C52 | C2×C26 | C2×C8 | C13 | C8 | C8 | C2×C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 6 | 4 | 6 | 6 | 6 | 12 | 12 | 48 |
Matrix representation of C52.4C8 ►in GL2(𝔽1249) generated by
| 156 | 0 |
| 316 | 1241 |
| 509 | 305 |
| 300 | 740 |
G:=sub<GL(2,GF(1249))| [156,316,0,1241],[509,300,305,740] >;
C52.4C8 in GAP, Magma, Sage, TeX
C_{52}._4C_8 % in TeX
G:=Group("C52.4C8"); // GroupNames label
G:=SmallGroup(416,19);
// by ID
G=gap.SmallGroup(416,19);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,217,50,69,13829]);
// Polycyclic
G:=Group<a,b|a^52=1,b^8=a^26,b*a*b^-1=a^-1>;
// generators/relations
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