metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C104.6C4, C52.34D4, C4.18D52, C8.1Dic13, C22.2Dic26, (C2×C8).5D13, (C2×C26).3Q8, (C2×C104).7C2, C52.56(C2×C4), (C2×C4).70D26, C26.14(C4⋊C4), C13⋊3(C8.C4), C4.8(C2×Dic13), C2.5(C52⋊3C4), C52.4C4.1C2, (C2×C52).97C22, SmallGroup(416,26)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C104.6C4
G = < a,b | a104=1, b4=a52, bab-1=a51 >
Smallest permutation representation of C104.6C4
►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)
(1 182 79 208 53 130 27 156)(2 129 80 155 54 181 28 207)(3 180 81 206 55 128 29 154)(4 127 82 153 56 179 30 205)(5 178 83 204 57 126 31 152)(6 125 84 151 58 177 32 203)(7 176 85 202 59 124 33 150)(8 123 86 149 60 175 34 201)(9 174 87 200 61 122 35 148)(10 121 88 147 62 173 36 199)(11 172 89 198 63 120 37 146)(12 119 90 145 64 171 38 197)(13 170 91 196 65 118 39 144)(14 117 92 143 66 169 40 195)(15 168 93 194 67 116 41 142)(16 115 94 141 68 167 42 193)(17 166 95 192 69 114 43 140)(18 113 96 139 70 165 44 191)(19 164 97 190 71 112 45 138)(20 111 98 137 72 163 46 189)(21 162 99 188 73 110 47 136)(22 109 100 135 74 161 48 187)(23 160 101 186 75 108 49 134)(24 107 102 133 76 159 50 185)(25 158 103 184 77 106 51 132)(26 105 104 131 78 157 52 183)
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), (1,182,79,208,53,130,27,156)(2,129,80,155,54,181,28,207)(3,180,81,206,55,128,29,154)(4,127,82,153,56,179,30,205)(5,178,83,204,57,126,31,152)(6,125,84,151,58,177,32,203)(7,176,85,202,59,124,33,150)(8,123,86,149,60,175,34,201)(9,174,87,200,61,122,35,148)(10,121,88,147,62,173,36,199)(11,172,89,198,63,120,37,146)(12,119,90,145,64,171,38,197)(13,170,91,196,65,118,39,144)(14,117,92,143,66,169,40,195)(15,168,93,194,67,116,41,142)(16,115,94,141,68,167,42,193)(17,166,95,192,69,114,43,140)(18,113,96,139,70,165,44,191)(19,164,97,190,71,112,45,138)(20,111,98,137,72,163,46,189)(21,162,99,188,73,110,47,136)(22,109,100,135,74,161,48,187)(23,160,101,186,75,108,49,134)(24,107,102,133,76,159,50,185)(25,158,103,184,77,106,51,132)(26,105,104,131,78,157,52,183)>;
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), (1,182,79,208,53,130,27,156)(2,129,80,155,54,181,28,207)(3,180,81,206,55,128,29,154)(4,127,82,153,56,179,30,205)(5,178,83,204,57,126,31,152)(6,125,84,151,58,177,32,203)(7,176,85,202,59,124,33,150)(8,123,86,149,60,175,34,201)(9,174,87,200,61,122,35,148)(10,121,88,147,62,173,36,199)(11,172,89,198,63,120,37,146)(12,119,90,145,64,171,38,197)(13,170,91,196,65,118,39,144)(14,117,92,143,66,169,40,195)(15,168,93,194,67,116,41,142)(16,115,94,141,68,167,42,193)(17,166,95,192,69,114,43,140)(18,113,96,139,70,165,44,191)(19,164,97,190,71,112,45,138)(20,111,98,137,72,163,46,189)(21,162,99,188,73,110,47,136)(22,109,100,135,74,161,48,187)(23,160,101,186,75,108,49,134)(24,107,102,133,76,159,50,185)(25,158,103,184,77,106,51,132)(26,105,104,131,78,157,52,183) );
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)], [(1,182,79,208,53,130,27,156),(2,129,80,155,54,181,28,207),(3,180,81,206,55,128,29,154),(4,127,82,153,56,179,30,205),(5,178,83,204,57,126,31,152),(6,125,84,151,58,177,32,203),(7,176,85,202,59,124,33,150),(8,123,86,149,60,175,34,201),(9,174,87,200,61,122,35,148),(10,121,88,147,62,173,36,199),(11,172,89,198,63,120,37,146),(12,119,90,145,64,171,38,197),(13,170,91,196,65,118,39,144),(14,117,92,143,66,169,40,195),(15,168,93,194,67,116,41,142),(16,115,94,141,68,167,42,193),(17,166,95,192,69,114,43,140),(18,113,96,139,70,165,44,191),(19,164,97,190,71,112,45,138),(20,111,98,137,72,163,46,189),(21,162,99,188,73,110,47,136),(22,109,100,135,74,161,48,187),(23,160,101,186,75,108,49,134),(24,107,102,133,76,159,50,185),(25,158,103,184,77,106,51,132),(26,105,104,131,78,157,52,183)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 13A | ··· | 13F | 26A | ··· | 26R | 52A | ··· | 52X | 104A | ··· | 104AV |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 52 | 52 | 52 | 52 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C4 | D4 | Q8 | D13 | C8.C4 | Dic13 | D26 | D52 | Dic26 | C104.6C4 |
| kernel | C104.6C4 | C52.4C4 | C2×C104 | C104 | C52 | C2×C26 | C2×C8 | C13 | C8 | C2×C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 6 | 4 | 12 | 6 | 12 | 12 | 48 |
Matrix representation of C104.6C4 ►in GL2(𝔽313) generated by
| 40 | 0 |
| 0 | 133 |
| 0 | 1 |
| 25 | 0 |
G:=sub<GL(2,GF(313))| [40,0,0,133],[0,25,1,0] >;
C104.6C4 in GAP, Magma, Sage, TeX
C_{104}._6C_4 % in TeX
G:=Group("C104.6C4"); // GroupNames label
G:=SmallGroup(416,26);
// by ID
G=gap.SmallGroup(416,26);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,55,86,579,69,13829]);
// Polycyclic
G:=Group<a,b|a^104=1,b^4=a^52,b*a*b^-1=a^51>;
// generators/relations
Export