metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C104.2C4, D26.1Q8, Dic13.10D4, C8.2(C13⋊C4), C26.3(C4⋊C4), C13⋊2C8.3C4, C52.10(C2×C4), (C8×D13).6C2, C13⋊1(C8.C4), C2.6(C52⋊C4), C52.C4.1C2, (C4×D13).27C22, C4.10(C2×C13⋊C4), SmallGroup(416,70)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C104.C4
G = < a,b | a104=1, b4=a52, bab-1=a83 >
Smallest permutation representation of C104.C4
►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)
(1 109 79 135 53 161 27 187)(2 208 104 114 54 156 52 166)(3 203 25 197 55 151 77 145)(4 198 50 176 56 146 102 124)(5 193 75 155 57 141 23 207)(6 188 100 134 58 136 48 186)(7 183 21 113 59 131 73 165)(8 178 46 196 60 126 98 144)(9 173 71 175 61 121 19 123)(10 168 96 154 62 116 44 206)(11 163 17 133 63 111 69 185)(12 158 42 112 64 106 94 164)(13 153 67 195 65 205 15 143)(14 148 92 174 66 200 40 122)(16 138 38 132 68 190 90 184)(18 128 88 194 70 180 36 142)(20 118 34 152 72 170 86 204)(22 108 84 110 74 160 32 162)(24 202 30 172 76 150 82 120)(26 192 80 130 78 140 28 182)(29 177 51 171 81 125 103 119)(31 167 101 129 83 115 49 181)(33 157 47 191 85 105 99 139)(35 147 97 149 87 199 45 201)(37 137 43 107 89 189 95 159)(39 127 93 169 91 179 41 117)
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), (1,109,79,135,53,161,27,187)(2,208,104,114,54,156,52,166)(3,203,25,197,55,151,77,145)(4,198,50,176,56,146,102,124)(5,193,75,155,57,141,23,207)(6,188,100,134,58,136,48,186)(7,183,21,113,59,131,73,165)(8,178,46,196,60,126,98,144)(9,173,71,175,61,121,19,123)(10,168,96,154,62,116,44,206)(11,163,17,133,63,111,69,185)(12,158,42,112,64,106,94,164)(13,153,67,195,65,205,15,143)(14,148,92,174,66,200,40,122)(16,138,38,132,68,190,90,184)(18,128,88,194,70,180,36,142)(20,118,34,152,72,170,86,204)(22,108,84,110,74,160,32,162)(24,202,30,172,76,150,82,120)(26,192,80,130,78,140,28,182)(29,177,51,171,81,125,103,119)(31,167,101,129,83,115,49,181)(33,157,47,191,85,105,99,139)(35,147,97,149,87,199,45,201)(37,137,43,107,89,189,95,159)(39,127,93,169,91,179,41,117)>;
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), (1,109,79,135,53,161,27,187)(2,208,104,114,54,156,52,166)(3,203,25,197,55,151,77,145)(4,198,50,176,56,146,102,124)(5,193,75,155,57,141,23,207)(6,188,100,134,58,136,48,186)(7,183,21,113,59,131,73,165)(8,178,46,196,60,126,98,144)(9,173,71,175,61,121,19,123)(10,168,96,154,62,116,44,206)(11,163,17,133,63,111,69,185)(12,158,42,112,64,106,94,164)(13,153,67,195,65,205,15,143)(14,148,92,174,66,200,40,122)(16,138,38,132,68,190,90,184)(18,128,88,194,70,180,36,142)(20,118,34,152,72,170,86,204)(22,108,84,110,74,160,32,162)(24,202,30,172,76,150,82,120)(26,192,80,130,78,140,28,182)(29,177,51,171,81,125,103,119)(31,167,101,129,83,115,49,181)(33,157,47,191,85,105,99,139)(35,147,97,149,87,199,45,201)(37,137,43,107,89,189,95,159)(39,127,93,169,91,179,41,117) );
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)], [(1,109,79,135,53,161,27,187),(2,208,104,114,54,156,52,166),(3,203,25,197,55,151,77,145),(4,198,50,176,56,146,102,124),(5,193,75,155,57,141,23,207),(6,188,100,134,58,136,48,186),(7,183,21,113,59,131,73,165),(8,178,46,196,60,126,98,144),(9,173,71,175,61,121,19,123),(10,168,96,154,62,116,44,206),(11,163,17,133,63,111,69,185),(12,158,42,112,64,106,94,164),(13,153,67,195,65,205,15,143),(14,148,92,174,66,200,40,122),(16,138,38,132,68,190,90,184),(18,128,88,194,70,180,36,142),(20,118,34,152,72,170,86,204),(22,108,84,110,74,160,32,162),(24,202,30,172,76,150,82,120),(26,192,80,130,78,140,28,182),(29,177,51,171,81,125,103,119),(31,167,101,129,83,115,49,181),(33,157,47,191,85,105,99,139),(35,147,97,149,87,199,45,201),(37,137,43,107,89,189,95,159),(39,127,93,169,91,179,41,117)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 13A | 13B | 13C | 26A | 26B | 26C | 52A | ··· | 52F | 104A | ··· | 104L |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 13 | 13 | 13 | 26 | 26 | 26 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 26 | 2 | 13 | 13 | 2 | 2 | 26 | 26 | 52 | 52 | 52 | 52 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | + | |||||
| image | C1 | C2 | C2 | C4 | C4 | D4 | Q8 | C8.C4 | C13⋊C4 | C2×C13⋊C4 | C52⋊C4 | C104.C4 |
| kernel | C104.C4 | C8×D13 | C52.C4 | C13⋊2C8 | C104 | Dic13 | D26 | C13 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 4 | 3 | 3 | 6 | 12 |
Matrix representation of C104.C4 ►in GL4(𝔽313) generated by
| 23 | 225 | 209 | 137 |
| 102 | 97 | 126 | 153 |
| 294 | 35 | 301 | 148 |
| 248 | 125 | 210 | 163 |
| 7 | 213 | 105 | 223 |
| 272 | 217 | 288 | 65 |
| 282 | 156 | 148 | 10 |
| 52 | 46 | 307 | 254 |
G:=sub<GL(4,GF(313))| [23,102,294,248,225,97,35,125,209,126,301,210,137,153,148,163],[7,272,282,52,213,217,156,46,105,288,148,307,223,65,10,254] >;
C104.C4 in GAP, Magma, Sage, TeX
C_{104}.C_4 % in TeX
G:=Group("C104.C4"); // GroupNames label
G:=SmallGroup(416,70);
// by ID
G=gap.SmallGroup(416,70);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,55,86,579,69,9221,3473]);
// Polycyclic
G:=Group<a,b|a^104=1,b^4=a^52,b*a*b^-1=a^83>;
// generators/relations
Export