metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8.6D26, C52.5D4, C13⋊3SD32, Q16⋊1D13, C26.10D8, D104.2C2, C104.4C22, C13⋊2C16⋊3C2, (C13×Q16)⋊1C2, C2.6(D4⋊D13), C4.3(C13⋊D4), SmallGroup(416,35)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8.6D26
G = < a,b,c | a8=1, b26=a4, c2=a3, bab-1=a-1, ac=ca, cbc-1=a-1b25 >
Smallest permutation representation of C8.6D26
►On 208 points
Generators in S208
(1 154 164 59 27 128 190 85)(2 86 191 129 28 60 165 155)(3 156 166 61 29 130 192 87)(4 88 193 131 30 62 167 105)(5 106 168 63 31 132 194 89)(6 90 195 133 32 64 169 107)(7 108 170 65 33 134 196 91)(8 92 197 135 34 66 171 109)(9 110 172 67 35 136 198 93)(10 94 199 137 36 68 173 111)(11 112 174 69 37 138 200 95)(12 96 201 139 38 70 175 113)(13 114 176 71 39 140 202 97)(14 98 203 141 40 72 177 115)(15 116 178 73 41 142 204 99)(16 100 205 143 42 74 179 117)(17 118 180 75 43 144 206 101)(18 102 207 145 44 76 181 119)(19 120 182 77 45 146 208 103)(20 104 157 147 46 78 183 121)(21 122 184 79 47 148 158 53)(22 54 159 149 48 80 185 123)(23 124 186 81 49 150 160 55)(24 56 161 151 50 82 187 125)(25 126 188 83 51 152 162 57)(26 58 163 153 52 84 189 127)
(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 26 59 153 190 189 154 58 27 52 85 127 164 163 128 84)(2 83 129 162 165 126 86 51 28 57 155 188 191 152 60 25)(3 24 61 151 192 187 156 56 29 50 87 125 166 161 130 82)(4 81 131 160 167 124 88 49 30 55 105 186 193 150 62 23)(5 22 63 149 194 185 106 54 31 48 89 123 168 159 132 80)(6 79 133 158 169 122 90 47 32 53 107 184 195 148 64 21)(7 20 65 147 196 183 108 104 33 46 91 121 170 157 134 78)(8 77 135 208 171 120 92 45 34 103 109 182 197 146 66 19)(9 18 67 145 198 181 110 102 35 44 93 119 172 207 136 76)(10 75 137 206 173 118 94 43 36 101 111 180 199 144 68 17)(11 16 69 143 200 179 112 100 37 42 95 117 174 205 138 74)(12 73 139 204 175 116 96 41 38 99 113 178 201 142 70 15)(13 14 71 141 202 177 114 98 39 40 97 115 176 203 140 72)
G:=sub<Sym(208)| (1,154,164,59,27,128,190,85)(2,86,191,129,28,60,165,155)(3,156,166,61,29,130,192,87)(4,88,193,131,30,62,167,105)(5,106,168,63,31,132,194,89)(6,90,195,133,32,64,169,107)(7,108,170,65,33,134,196,91)(8,92,197,135,34,66,171,109)(9,110,172,67,35,136,198,93)(10,94,199,137,36,68,173,111)(11,112,174,69,37,138,200,95)(12,96,201,139,38,70,175,113)(13,114,176,71,39,140,202,97)(14,98,203,141,40,72,177,115)(15,116,178,73,41,142,204,99)(16,100,205,143,42,74,179,117)(17,118,180,75,43,144,206,101)(18,102,207,145,44,76,181,119)(19,120,182,77,45,146,208,103)(20,104,157,147,46,78,183,121)(21,122,184,79,47,148,158,53)(22,54,159,149,48,80,185,123)(23,124,186,81,49,150,160,55)(24,56,161,151,50,82,187,125)(25,126,188,83,51,152,162,57)(26,58,163,153,52,84,189,127), (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), (1,26,59,153,190,189,154,58,27,52,85,127,164,163,128,84)(2,83,129,162,165,126,86,51,28,57,155,188,191,152,60,25)(3,24,61,151,192,187,156,56,29,50,87,125,166,161,130,82)(4,81,131,160,167,124,88,49,30,55,105,186,193,150,62,23)(5,22,63,149,194,185,106,54,31,48,89,123,168,159,132,80)(6,79,133,158,169,122,90,47,32,53,107,184,195,148,64,21)(7,20,65,147,196,183,108,104,33,46,91,121,170,157,134,78)(8,77,135,208,171,120,92,45,34,103,109,182,197,146,66,19)(9,18,67,145,198,181,110,102,35,44,93,119,172,207,136,76)(10,75,137,206,173,118,94,43,36,101,111,180,199,144,68,17)(11,16,69,143,200,179,112,100,37,42,95,117,174,205,138,74)(12,73,139,204,175,116,96,41,38,99,113,178,201,142,70,15)(13,14,71,141,202,177,114,98,39,40,97,115,176,203,140,72)>;
G:=Group( (1,154,164,59,27,128,190,85)(2,86,191,129,28,60,165,155)(3,156,166,61,29,130,192,87)(4,88,193,131,30,62,167,105)(5,106,168,63,31,132,194,89)(6,90,195,133,32,64,169,107)(7,108,170,65,33,134,196,91)(8,92,197,135,34,66,171,109)(9,110,172,67,35,136,198,93)(10,94,199,137,36,68,173,111)(11,112,174,69,37,138,200,95)(12,96,201,139,38,70,175,113)(13,114,176,71,39,140,202,97)(14,98,203,141,40,72,177,115)(15,116,178,73,41,142,204,99)(16,100,205,143,42,74,179,117)(17,118,180,75,43,144,206,101)(18,102,207,145,44,76,181,119)(19,120,182,77,45,146,208,103)(20,104,157,147,46,78,183,121)(21,122,184,79,47,148,158,53)(22,54,159,149,48,80,185,123)(23,124,186,81,49,150,160,55)(24,56,161,151,50,82,187,125)(25,126,188,83,51,152,162,57)(26,58,163,153,52,84,189,127), (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), (1,26,59,153,190,189,154,58,27,52,85,127,164,163,128,84)(2,83,129,162,165,126,86,51,28,57,155,188,191,152,60,25)(3,24,61,151,192,187,156,56,29,50,87,125,166,161,130,82)(4,81,131,160,167,124,88,49,30,55,105,186,193,150,62,23)(5,22,63,149,194,185,106,54,31,48,89,123,168,159,132,80)(6,79,133,158,169,122,90,47,32,53,107,184,195,148,64,21)(7,20,65,147,196,183,108,104,33,46,91,121,170,157,134,78)(8,77,135,208,171,120,92,45,34,103,109,182,197,146,66,19)(9,18,67,145,198,181,110,102,35,44,93,119,172,207,136,76)(10,75,137,206,173,118,94,43,36,101,111,180,199,144,68,17)(11,16,69,143,200,179,112,100,37,42,95,117,174,205,138,74)(12,73,139,204,175,116,96,41,38,99,113,178,201,142,70,15)(13,14,71,141,202,177,114,98,39,40,97,115,176,203,140,72) );
G=PermutationGroup([[(1,154,164,59,27,128,190,85),(2,86,191,129,28,60,165,155),(3,156,166,61,29,130,192,87),(4,88,193,131,30,62,167,105),(5,106,168,63,31,132,194,89),(6,90,195,133,32,64,169,107),(7,108,170,65,33,134,196,91),(8,92,197,135,34,66,171,109),(9,110,172,67,35,136,198,93),(10,94,199,137,36,68,173,111),(11,112,174,69,37,138,200,95),(12,96,201,139,38,70,175,113),(13,114,176,71,39,140,202,97),(14,98,203,141,40,72,177,115),(15,116,178,73,41,142,204,99),(16,100,205,143,42,74,179,117),(17,118,180,75,43,144,206,101),(18,102,207,145,44,76,181,119),(19,120,182,77,45,146,208,103),(20,104,157,147,46,78,183,121),(21,122,184,79,47,148,158,53),(22,54,159,149,48,80,185,123),(23,124,186,81,49,150,160,55),(24,56,161,151,50,82,187,125),(25,126,188,83,51,152,162,57),(26,58,163,153,52,84,189,127)], [(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)], [(1,26,59,153,190,189,154,58,27,52,85,127,164,163,128,84),(2,83,129,162,165,126,86,51,28,57,155,188,191,152,60,25),(3,24,61,151,192,187,156,56,29,50,87,125,166,161,130,82),(4,81,131,160,167,124,88,49,30,55,105,186,193,150,62,23),(5,22,63,149,194,185,106,54,31,48,89,123,168,159,132,80),(6,79,133,158,169,122,90,47,32,53,107,184,195,148,64,21),(7,20,65,147,196,183,108,104,33,46,91,121,170,157,134,78),(8,77,135,208,171,120,92,45,34,103,109,182,197,146,66,19),(9,18,67,145,198,181,110,102,35,44,93,119,172,207,136,76),(10,75,137,206,173,118,94,43,36,101,111,180,199,144,68,17),(11,16,69,143,200,179,112,100,37,42,95,117,174,205,138,74),(12,73,139,204,175,116,96,41,38,99,113,178,201,142,70,15),(13,14,71,141,202,177,114,98,39,40,97,115,176,203,140,72)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 8A | 8B | 13A | ··· | 13F | 16A | 16B | 16C | 16D | 26A | ··· | 26F | 52A | ··· | 52F | 52G | ··· | 52R | 104A | ··· | 104L |
| order | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 13 | ··· | 13 | 16 | 16 | 16 | 16 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 104 | 2 | 8 | 2 | 2 | 2 | ··· | 2 | 26 | 26 | 26 | 26 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | D4 | D8 | D13 | SD32 | D26 | C13⋊D4 | D4⋊D13 | C8.6D26 |
| kernel | C8.6D26 | C13⋊2C16 | D104 | C13×Q16 | C52 | C26 | Q16 | C13 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 6 | 4 | 6 | 12 | 6 | 12 |
Matrix representation of C8.6D26 ►in GL4(𝔽1249) generated by
| 1248 | 0 | 0 | 0 |
| 0 | 1248 | 0 | 0 |
| 0 | 0 | 1165 | 1152 |
| 0 | 0 | 850 | 134 |
| 384 | 217 | 0 | 0 |
| 18 | 111 | 0 | 0 |
| 0 | 0 | 785 | 165 |
| 0 | 0 | 686 | 464 |
| 157 | 537 | 0 | 0 |
| 817 | 1092 | 0 | 0 |
| 0 | 0 | 1161 | 1084 |
| 0 | 0 | 184 | 785 |
G:=sub<GL(4,GF(1249))| [1248,0,0,0,0,1248,0,0,0,0,1165,850,0,0,1152,134],[384,18,0,0,217,111,0,0,0,0,785,686,0,0,165,464],[157,817,0,0,537,1092,0,0,0,0,1161,184,0,0,1084,785] >;
C8.6D26 in GAP, Magma, Sage, TeX
C_8._6D_{26} % in TeX
G:=Group("C8.6D26"); // GroupNames label
G:=SmallGroup(416,35);
// by ID
G=gap.SmallGroup(416,35);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,73,103,218,116,122,579,297,69,13829]);
// Polycyclic
G:=Group<a,b,c|a^8=1,b^26=a^4,c^2=a^3,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^-1*b^25>;
// generators/relations
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