metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8.1D26, C4.15D52, C52.13D4, Dic52⋊2C2, C22.6D52, M4(2)⋊2D13, C104.1C22, C52.33C23, D52.8C22, Dic26.8C22, (C2×C26).6D4, C104⋊C2⋊2C2, C2.16(C2×D52), (C2×C4).16D26, C26.14(C2×D4), (C2×Dic26)⋊8C2, C13⋊1(C8.C22), D52⋊5C2.4C2, (C13×M4(2))⋊2C2, (C2×C52).28C22, C4.31(C22×D13), SmallGroup(416,130)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8.D26
G = < a,b,c | a8=1, b26=c2=a4, bab-1=a5, cac-1=a-1, cbc-1=b25 >
Subgroups: 456 in 60 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C13, M4(2), SD16, Q16, C2×Q8, C4○D4, D13, C26, C26, C8.C22, Dic13, C52, D26, C2×C26, C104, Dic26, Dic26, Dic26, C4×D13, D52, C2×Dic13, C13⋊D4, C2×C52, C104⋊C2, Dic52, C13×M4(2), C2×Dic26, D52⋊5C2, C8.D26
Quotients: C1, C2, C22, D4, C23, C2×D4, D13, C8.C22, D26, D52, C22×D13, C2×D52, C8.D26
►On 208 points
Generators in S208
(1 72 40 85 27 98 14 59)(2 99 41 60 28 73 15 86)(3 74 42 87 29 100 16 61)(4 101 43 62 30 75 17 88)(5 76 44 89 31 102 18 63)(6 103 45 64 32 77 19 90)(7 78 46 91 33 104 20 65)(8 53 47 66 34 79 21 92)(9 80 48 93 35 54 22 67)(10 55 49 68 36 81 23 94)(11 82 50 95 37 56 24 69)(12 57 51 70 38 83 25 96)(13 84 52 97 39 58 26 71)(105 205 118 192 131 179 144 166)(106 180 119 167 132 206 145 193)(107 207 120 194 133 181 146 168)(108 182 121 169 134 208 147 195)(109 157 122 196 135 183 148 170)(110 184 123 171 136 158 149 197)(111 159 124 198 137 185 150 172)(112 186 125 173 138 160 151 199)(113 161 126 200 139 187 152 174)(114 188 127 175 140 162 153 201)(115 163 128 202 141 189 154 176)(116 190 129 177 142 164 155 203)(117 165 130 204 143 191 156 178)
(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)
(1 132 27 106)(2 105 28 131)(3 130 29 156)(4 155 30 129)(5 128 31 154)(6 153 32 127)(7 126 33 152)(8 151 34 125)(9 124 35 150)(10 149 36 123)(11 122 37 148)(12 147 38 121)(13 120 39 146)(14 145 40 119)(15 118 41 144)(16 143 42 117)(17 116 43 142)(18 141 44 115)(19 114 45 140)(20 139 46 113)(21 112 47 138)(22 137 48 111)(23 110 49 136)(24 135 50 109)(25 108 51 134)(26 133 52 107)(53 160 79 186)(54 185 80 159)(55 158 81 184)(56 183 82 157)(57 208 83 182)(58 181 84 207)(59 206 85 180)(60 179 86 205)(61 204 87 178)(62 177 88 203)(63 202 89 176)(64 175 90 201)(65 200 91 174)(66 173 92 199)(67 198 93 172)(68 171 94 197)(69 196 95 170)(70 169 96 195)(71 194 97 168)(72 167 98 193)(73 192 99 166)(74 165 100 191)(75 190 101 164)(76 163 102 189)(77 188 103 162)(78 161 104 187)
G:=sub<Sym(208)| (1,72,40,85,27,98,14,59)(2,99,41,60,28,73,15,86)(3,74,42,87,29,100,16,61)(4,101,43,62,30,75,17,88)(5,76,44,89,31,102,18,63)(6,103,45,64,32,77,19,90)(7,78,46,91,33,104,20,65)(8,53,47,66,34,79,21,92)(9,80,48,93,35,54,22,67)(10,55,49,68,36,81,23,94)(11,82,50,95,37,56,24,69)(12,57,51,70,38,83,25,96)(13,84,52,97,39,58,26,71)(105,205,118,192,131,179,144,166)(106,180,119,167,132,206,145,193)(107,207,120,194,133,181,146,168)(108,182,121,169,134,208,147,195)(109,157,122,196,135,183,148,170)(110,184,123,171,136,158,149,197)(111,159,124,198,137,185,150,172)(112,186,125,173,138,160,151,199)(113,161,126,200,139,187,152,174)(114,188,127,175,140,162,153,201)(115,163,128,202,141,189,154,176)(116,190,129,177,142,164,155,203)(117,165,130,204,143,191,156,178), (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), (1,132,27,106)(2,105,28,131)(3,130,29,156)(4,155,30,129)(5,128,31,154)(6,153,32,127)(7,126,33,152)(8,151,34,125)(9,124,35,150)(10,149,36,123)(11,122,37,148)(12,147,38,121)(13,120,39,146)(14,145,40,119)(15,118,41,144)(16,143,42,117)(17,116,43,142)(18,141,44,115)(19,114,45,140)(20,139,46,113)(21,112,47,138)(22,137,48,111)(23,110,49,136)(24,135,50,109)(25,108,51,134)(26,133,52,107)(53,160,79,186)(54,185,80,159)(55,158,81,184)(56,183,82,157)(57,208,83,182)(58,181,84,207)(59,206,85,180)(60,179,86,205)(61,204,87,178)(62,177,88,203)(63,202,89,176)(64,175,90,201)(65,200,91,174)(66,173,92,199)(67,198,93,172)(68,171,94,197)(69,196,95,170)(70,169,96,195)(71,194,97,168)(72,167,98,193)(73,192,99,166)(74,165,100,191)(75,190,101,164)(76,163,102,189)(77,188,103,162)(78,161,104,187)>;
G:=Group( (1,72,40,85,27,98,14,59)(2,99,41,60,28,73,15,86)(3,74,42,87,29,100,16,61)(4,101,43,62,30,75,17,88)(5,76,44,89,31,102,18,63)(6,103,45,64,32,77,19,90)(7,78,46,91,33,104,20,65)(8,53,47,66,34,79,21,92)(9,80,48,93,35,54,22,67)(10,55,49,68,36,81,23,94)(11,82,50,95,37,56,24,69)(12,57,51,70,38,83,25,96)(13,84,52,97,39,58,26,71)(105,205,118,192,131,179,144,166)(106,180,119,167,132,206,145,193)(107,207,120,194,133,181,146,168)(108,182,121,169,134,208,147,195)(109,157,122,196,135,183,148,170)(110,184,123,171,136,158,149,197)(111,159,124,198,137,185,150,172)(112,186,125,173,138,160,151,199)(113,161,126,200,139,187,152,174)(114,188,127,175,140,162,153,201)(115,163,128,202,141,189,154,176)(116,190,129,177,142,164,155,203)(117,165,130,204,143,191,156,178), (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), (1,132,27,106)(2,105,28,131)(3,130,29,156)(4,155,30,129)(5,128,31,154)(6,153,32,127)(7,126,33,152)(8,151,34,125)(9,124,35,150)(10,149,36,123)(11,122,37,148)(12,147,38,121)(13,120,39,146)(14,145,40,119)(15,118,41,144)(16,143,42,117)(17,116,43,142)(18,141,44,115)(19,114,45,140)(20,139,46,113)(21,112,47,138)(22,137,48,111)(23,110,49,136)(24,135,50,109)(25,108,51,134)(26,133,52,107)(53,160,79,186)(54,185,80,159)(55,158,81,184)(56,183,82,157)(57,208,83,182)(58,181,84,207)(59,206,85,180)(60,179,86,205)(61,204,87,178)(62,177,88,203)(63,202,89,176)(64,175,90,201)(65,200,91,174)(66,173,92,199)(67,198,93,172)(68,171,94,197)(69,196,95,170)(70,169,96,195)(71,194,97,168)(72,167,98,193)(73,192,99,166)(74,165,100,191)(75,190,101,164)(76,163,102,189)(77,188,103,162)(78,161,104,187) );
G=PermutationGroup([[(1,72,40,85,27,98,14,59),(2,99,41,60,28,73,15,86),(3,74,42,87,29,100,16,61),(4,101,43,62,30,75,17,88),(5,76,44,89,31,102,18,63),(6,103,45,64,32,77,19,90),(7,78,46,91,33,104,20,65),(8,53,47,66,34,79,21,92),(9,80,48,93,35,54,22,67),(10,55,49,68,36,81,23,94),(11,82,50,95,37,56,24,69),(12,57,51,70,38,83,25,96),(13,84,52,97,39,58,26,71),(105,205,118,192,131,179,144,166),(106,180,119,167,132,206,145,193),(107,207,120,194,133,181,146,168),(108,182,121,169,134,208,147,195),(109,157,122,196,135,183,148,170),(110,184,123,171,136,158,149,197),(111,159,124,198,137,185,150,172),(112,186,125,173,138,160,151,199),(113,161,126,200,139,187,152,174),(114,188,127,175,140,162,153,201),(115,163,128,202,141,189,154,176),(116,190,129,177,142,164,155,203),(117,165,130,204,143,191,156,178)], [(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)], [(1,132,27,106),(2,105,28,131),(3,130,29,156),(4,155,30,129),(5,128,31,154),(6,153,32,127),(7,126,33,152),(8,151,34,125),(9,124,35,150),(10,149,36,123),(11,122,37,148),(12,147,38,121),(13,120,39,146),(14,145,40,119),(15,118,41,144),(16,143,42,117),(17,116,43,142),(18,141,44,115),(19,114,45,140),(20,139,46,113),(21,112,47,138),(22,137,48,111),(23,110,49,136),(24,135,50,109),(25,108,51,134),(26,133,52,107),(53,160,79,186),(54,185,80,159),(55,158,81,184),(56,183,82,157),(57,208,83,182),(58,181,84,207),(59,206,85,180),(60,179,86,205),(61,204,87,178),(62,177,88,203),(63,202,89,176),(64,175,90,201),(65,200,91,174),(66,173,92,199),(67,198,93,172),(68,171,94,197),(69,196,95,170),(70,169,96,195),(71,194,97,168),(72,167,98,193),(73,192,99,166),(74,165,100,191),(75,190,101,164),(76,163,102,189),(77,188,103,162),(78,161,104,187)]])
71 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 8A | 8B | 13A | ··· | 13F | 26A | ··· | 26F | 26G | ··· | 26L | 52A | ··· | 52L | 52M | ··· | 52R | 104A | ··· | 104X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 2 | 52 | 2 | 2 | 52 | 52 | 52 | 4 | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
71 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D13 | D26 | D26 | D52 | D52 | C8.C22 | C8.D26 |
| kernel | C8.D26 | C104⋊C2 | Dic52 | C13×M4(2) | C2×Dic26 | D52⋊5C2 | C52 | C2×C26 | M4(2) | C8 | C2×C4 | C4 | C22 | C13 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 6 | 12 | 6 | 12 | 12 | 1 | 12 |
Matrix representation of C8.D26 ►in GL4(𝔽313) generated by
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 228 | 271 | 0 | 0 |
| 23 | 85 | 0 | 0 |
| 187 | 206 | 0 | 0 |
| 230 | 9 | 0 | 0 |
| 0 | 0 | 126 | 107 |
| 0 | 0 | 83 | 304 |
| 0 | 0 | 284 | 268 |
| 0 | 0 | 130 | 29 |
| 284 | 268 | 0 | 0 |
| 130 | 29 | 0 | 0 |
G:=sub<GL(4,GF(313))| [0,0,228,23,0,0,271,85,1,0,0,0,0,1,0,0],[187,230,0,0,206,9,0,0,0,0,126,83,0,0,107,304],[0,0,284,130,0,0,268,29,284,130,0,0,268,29,0,0] >;
C8.D26 in GAP, Magma, Sage, TeX
C_8.D_{26} % in TeX
G:=Group("C8.D26"); // GroupNames label
G:=SmallGroup(416,130);
// by ID
G=gap.SmallGroup(416,130);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,103,218,188,50,579,69,13829]);
// Polycyclic
G:=Group<a,b,c|a^8=1,b^26=c^2=a^4,b*a*b^-1=a^5,c*a*c^-1=a^-1,c*b*c^-1=b^25>;
// generators/relations