metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C104.1C4, D26.2Q8, Dic13.11D4, C8.1(C13⋊C4), C26.4(C4⋊C4), C13⋊2C8.4C4, C52.11(C2×C4), (C8×D13).4C2, C13⋊2(C8.C4), C2.7(C52⋊C4), C52.C4.2C2, (C4×D13).28C22, C4.11(C2×C13⋊C4), SmallGroup(416,71)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C104.1C4
G = < a,b | a104=1, b4=a52, bab-1=a31 >
Smallest permutation representation of C104.1C4
►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)
(1 147 27 121 53 199 79 173)(2 194 52 152 54 142 104 204)(3 137 77 183 55 189 25 131)(4 184 102 110 56 132 50 162)(5 127 23 141 57 179 75 193)(6 174 48 172 58 122 100 120)(7 117 73 203 59 169 21 151)(8 164 98 130 60 112 46 182)(9 107 19 161 61 159 71 109)(10 154 44 192 62 206 96 140)(11 201 69 119 63 149 17 171)(12 144 94 150 64 196 42 202)(13 191 15 181 65 139 67 129)(14 134 40 108 66 186 92 160)(16 124 90 170 68 176 38 118)(18 114 36 128 70 166 88 180)(20 208 86 190 72 156 34 138)(22 198 32 148 74 146 84 200)(24 188 82 106 76 136 30 158)(26 178 28 168 78 126 80 116)(29 111 103 157 81 163 51 105)(31 205 49 115 83 153 101 167)(33 195 99 177 85 143 47 125)(35 185 45 135 87 133 97 187)(37 175 95 197 89 123 43 145)(39 165 41 155 91 113 93 207)
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), (1,147,27,121,53,199,79,173)(2,194,52,152,54,142,104,204)(3,137,77,183,55,189,25,131)(4,184,102,110,56,132,50,162)(5,127,23,141,57,179,75,193)(6,174,48,172,58,122,100,120)(7,117,73,203,59,169,21,151)(8,164,98,130,60,112,46,182)(9,107,19,161,61,159,71,109)(10,154,44,192,62,206,96,140)(11,201,69,119,63,149,17,171)(12,144,94,150,64,196,42,202)(13,191,15,181,65,139,67,129)(14,134,40,108,66,186,92,160)(16,124,90,170,68,176,38,118)(18,114,36,128,70,166,88,180)(20,208,86,190,72,156,34,138)(22,198,32,148,74,146,84,200)(24,188,82,106,76,136,30,158)(26,178,28,168,78,126,80,116)(29,111,103,157,81,163,51,105)(31,205,49,115,83,153,101,167)(33,195,99,177,85,143,47,125)(35,185,45,135,87,133,97,187)(37,175,95,197,89,123,43,145)(39,165,41,155,91,113,93,207)>;
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), (1,147,27,121,53,199,79,173)(2,194,52,152,54,142,104,204)(3,137,77,183,55,189,25,131)(4,184,102,110,56,132,50,162)(5,127,23,141,57,179,75,193)(6,174,48,172,58,122,100,120)(7,117,73,203,59,169,21,151)(8,164,98,130,60,112,46,182)(9,107,19,161,61,159,71,109)(10,154,44,192,62,206,96,140)(11,201,69,119,63,149,17,171)(12,144,94,150,64,196,42,202)(13,191,15,181,65,139,67,129)(14,134,40,108,66,186,92,160)(16,124,90,170,68,176,38,118)(18,114,36,128,70,166,88,180)(20,208,86,190,72,156,34,138)(22,198,32,148,74,146,84,200)(24,188,82,106,76,136,30,158)(26,178,28,168,78,126,80,116)(29,111,103,157,81,163,51,105)(31,205,49,115,83,153,101,167)(33,195,99,177,85,143,47,125)(35,185,45,135,87,133,97,187)(37,175,95,197,89,123,43,145)(39,165,41,155,91,113,93,207) );
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)], [(1,147,27,121,53,199,79,173),(2,194,52,152,54,142,104,204),(3,137,77,183,55,189,25,131),(4,184,102,110,56,132,50,162),(5,127,23,141,57,179,75,193),(6,174,48,172,58,122,100,120),(7,117,73,203,59,169,21,151),(8,164,98,130,60,112,46,182),(9,107,19,161,61,159,71,109),(10,154,44,192,62,206,96,140),(11,201,69,119,63,149,17,171),(12,144,94,150,64,196,42,202),(13,191,15,181,65,139,67,129),(14,134,40,108,66,186,92,160),(16,124,90,170,68,176,38,118),(18,114,36,128,70,166,88,180),(20,208,86,190,72,156,34,138),(22,198,32,148,74,146,84,200),(24,188,82,106,76,136,30,158),(26,178,28,168,78,126,80,116),(29,111,103,157,81,163,51,105),(31,205,49,115,83,153,101,167),(33,195,99,177,85,143,47,125),(35,185,45,135,87,133,97,187),(37,175,95,197,89,123,43,145),(39,165,41,155,91,113,93,207)]])
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 | C104.1C4 |
| kernel | C104.1C4 | 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 C104.1C4 ►in GL4(𝔽313) generated by
| 191 | 122 | 251 | 307 |
| 42 | 211 | 40 | 132 |
| 42 | 88 | 85 | 48 |
| 191 | 62 | 6 | 279 |
| 213 | 120 | 116 | 221 |
| 38 | 177 | 72 | 250 |
| 241 | 136 | 275 | 109 |
| 197 | 193 | 100 | 274 |
G:=sub<GL(4,GF(313))| [191,42,42,191,122,211,88,62,251,40,85,6,307,132,48,279],[213,38,241,197,120,177,136,193,116,72,275,100,221,250,109,274] >;
C104.1C4 in GAP, Magma, Sage, TeX
C_{104}._1C_4 % in TeX
G:=Group("C104.1C4"); // GroupNames label
G:=SmallGroup(416,71);
// by ID
G=gap.SmallGroup(416,71);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,151,86,579,69,9221,3473]);
// Polycyclic
G:=Group<a,b|a^104=1,b^4=a^52,b*a*b^-1=a^31>;
// generators/relations
Export