Copied to
clipboard

## G = C8.D26order 416 = 25·13

### 1st non-split extension by C8 of D26 acting via D26/C13=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C8.D26
 Chief series C1 — C13 — C26 — C52 — D52 — D52⋊5C2 — C8.D26
 Lower central C13 — C26 — C52 — C8.D26
 Upper central C1 — C2 — C2×C4 — M4(2)

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 [×2], C4 [×2], C4 [×3], C22, C22, C8 [×2], C2×C4, C2×C4 [×2], D4 [×2], Q8 [×4], C13, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, D13, C26, C26, C8.C22, Dic13 [×3], C52 [×2], D26, C2×C26, C104 [×2], Dic26, Dic26 [×2], Dic26, C4×D13, D52, C2×Dic13, C13⋊D4, C2×C52, C104⋊C2 [×2], Dic52 [×2], C13×M4(2), C2×Dic26, D525C2, C8.D26
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C2×D4, D13, C8.C22, D26 [×3], D52 [×2], C22×D13, C2×D52, C8.D26

Smallest permutation representation of C8.D26
On 208 points
Generators in S208
```(1 102 40 63 27 76 14 89)(2 77 41 90 28 103 15 64)(3 104 42 65 29 78 16 91)(4 79 43 92 30 53 17 66)(5 54 44 67 31 80 18 93)(6 81 45 94 32 55 19 68)(7 56 46 69 33 82 20 95)(8 83 47 96 34 57 21 70)(9 58 48 71 35 84 22 97)(10 85 49 98 36 59 23 72)(11 60 50 73 37 86 24 99)(12 87 51 100 38 61 25 74)(13 62 52 75 39 88 26 101)(105 167 118 206 131 193 144 180)(106 194 119 181 132 168 145 207)(107 169 120 208 133 195 146 182)(108 196 121 183 134 170 147 157)(109 171 122 158 135 197 148 184)(110 198 123 185 136 172 149 159)(111 173 124 160 137 199 150 186)(112 200 125 187 138 174 151 161)(113 175 126 162 139 201 152 188)(114 202 127 189 140 176 153 163)(115 177 128 164 141 203 154 190)(116 204 129 191 142 178 155 165)(117 179 130 166 143 205 156 192)
(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)
(1 116 27 142)(2 141 28 115)(3 114 29 140)(4 139 30 113)(5 112 31 138)(6 137 32 111)(7 110 33 136)(8 135 34 109)(9 108 35 134)(10 133 36 107)(11 106 37 132)(12 131 38 105)(13 156 39 130)(14 129 40 155)(15 154 41 128)(16 127 42 153)(17 152 43 126)(18 125 44 151)(19 150 45 124)(20 123 46 149)(21 148 47 122)(22 121 48 147)(23 146 49 120)(24 119 50 145)(25 144 51 118)(26 117 52 143)(53 188 79 162)(54 161 80 187)(55 186 81 160)(56 159 82 185)(57 184 83 158)(58 157 84 183)(59 182 85 208)(60 207 86 181)(61 180 87 206)(62 205 88 179)(63 178 89 204)(64 203 90 177)(65 176 91 202)(66 201 92 175)(67 174 93 200)(68 199 94 173)(69 172 95 198)(70 197 96 171)(71 170 97 196)(72 195 98 169)(73 168 99 194)(74 193 100 167)(75 166 101 192)(76 191 102 165)(77 164 103 190)(78 189 104 163)```

`G:=sub<Sym(208)| (1,102,40,63,27,76,14,89)(2,77,41,90,28,103,15,64)(3,104,42,65,29,78,16,91)(4,79,43,92,30,53,17,66)(5,54,44,67,31,80,18,93)(6,81,45,94,32,55,19,68)(7,56,46,69,33,82,20,95)(8,83,47,96,34,57,21,70)(9,58,48,71,35,84,22,97)(10,85,49,98,36,59,23,72)(11,60,50,73,37,86,24,99)(12,87,51,100,38,61,25,74)(13,62,52,75,39,88,26,101)(105,167,118,206,131,193,144,180)(106,194,119,181,132,168,145,207)(107,169,120,208,133,195,146,182)(108,196,121,183,134,170,147,157)(109,171,122,158,135,197,148,184)(110,198,123,185,136,172,149,159)(111,173,124,160,137,199,150,186)(112,200,125,187,138,174,151,161)(113,175,126,162,139,201,152,188)(114,202,127,189,140,176,153,163)(115,177,128,164,141,203,154,190)(116,204,129,191,142,178,155,165)(117,179,130,166,143,205,156,192), (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), (1,116,27,142)(2,141,28,115)(3,114,29,140)(4,139,30,113)(5,112,31,138)(6,137,32,111)(7,110,33,136)(8,135,34,109)(9,108,35,134)(10,133,36,107)(11,106,37,132)(12,131,38,105)(13,156,39,130)(14,129,40,155)(15,154,41,128)(16,127,42,153)(17,152,43,126)(18,125,44,151)(19,150,45,124)(20,123,46,149)(21,148,47,122)(22,121,48,147)(23,146,49,120)(24,119,50,145)(25,144,51,118)(26,117,52,143)(53,188,79,162)(54,161,80,187)(55,186,81,160)(56,159,82,185)(57,184,83,158)(58,157,84,183)(59,182,85,208)(60,207,86,181)(61,180,87,206)(62,205,88,179)(63,178,89,204)(64,203,90,177)(65,176,91,202)(66,201,92,175)(67,174,93,200)(68,199,94,173)(69,172,95,198)(70,197,96,171)(71,170,97,196)(72,195,98,169)(73,168,99,194)(74,193,100,167)(75,166,101,192)(76,191,102,165)(77,164,103,190)(78,189,104,163)>;`

`G:=Group( (1,102,40,63,27,76,14,89)(2,77,41,90,28,103,15,64)(3,104,42,65,29,78,16,91)(4,79,43,92,30,53,17,66)(5,54,44,67,31,80,18,93)(6,81,45,94,32,55,19,68)(7,56,46,69,33,82,20,95)(8,83,47,96,34,57,21,70)(9,58,48,71,35,84,22,97)(10,85,49,98,36,59,23,72)(11,60,50,73,37,86,24,99)(12,87,51,100,38,61,25,74)(13,62,52,75,39,88,26,101)(105,167,118,206,131,193,144,180)(106,194,119,181,132,168,145,207)(107,169,120,208,133,195,146,182)(108,196,121,183,134,170,147,157)(109,171,122,158,135,197,148,184)(110,198,123,185,136,172,149,159)(111,173,124,160,137,199,150,186)(112,200,125,187,138,174,151,161)(113,175,126,162,139,201,152,188)(114,202,127,189,140,176,153,163)(115,177,128,164,141,203,154,190)(116,204,129,191,142,178,155,165)(117,179,130,166,143,205,156,192), (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), (1,116,27,142)(2,141,28,115)(3,114,29,140)(4,139,30,113)(5,112,31,138)(6,137,32,111)(7,110,33,136)(8,135,34,109)(9,108,35,134)(10,133,36,107)(11,106,37,132)(12,131,38,105)(13,156,39,130)(14,129,40,155)(15,154,41,128)(16,127,42,153)(17,152,43,126)(18,125,44,151)(19,150,45,124)(20,123,46,149)(21,148,47,122)(22,121,48,147)(23,146,49,120)(24,119,50,145)(25,144,51,118)(26,117,52,143)(53,188,79,162)(54,161,80,187)(55,186,81,160)(56,159,82,185)(57,184,83,158)(58,157,84,183)(59,182,85,208)(60,207,86,181)(61,180,87,206)(62,205,88,179)(63,178,89,204)(64,203,90,177)(65,176,91,202)(66,201,92,175)(67,174,93,200)(68,199,94,173)(69,172,95,198)(70,197,96,171)(71,170,97,196)(72,195,98,169)(73,168,99,194)(74,193,100,167)(75,166,101,192)(76,191,102,165)(77,164,103,190)(78,189,104,163) );`

`G=PermutationGroup([(1,102,40,63,27,76,14,89),(2,77,41,90,28,103,15,64),(3,104,42,65,29,78,16,91),(4,79,43,92,30,53,17,66),(5,54,44,67,31,80,18,93),(6,81,45,94,32,55,19,68),(7,56,46,69,33,82,20,95),(8,83,47,96,34,57,21,70),(9,58,48,71,35,84,22,97),(10,85,49,98,36,59,23,72),(11,60,50,73,37,86,24,99),(12,87,51,100,38,61,25,74),(13,62,52,75,39,88,26,101),(105,167,118,206,131,193,144,180),(106,194,119,181,132,168,145,207),(107,169,120,208,133,195,146,182),(108,196,121,183,134,170,147,157),(109,171,122,158,135,197,148,184),(110,198,123,185,136,172,149,159),(111,173,124,160,137,199,150,186),(112,200,125,187,138,174,151,161),(113,175,126,162,139,201,152,188),(114,202,127,189,140,176,153,163),(115,177,128,164,141,203,154,190),(116,204,129,191,142,178,155,165),(117,179,130,166,143,205,156,192)], [(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)], [(1,116,27,142),(2,141,28,115),(3,114,29,140),(4,139,30,113),(5,112,31,138),(6,137,32,111),(7,110,33,136),(8,135,34,109),(9,108,35,134),(10,133,36,107),(11,106,37,132),(12,131,38,105),(13,156,39,130),(14,129,40,155),(15,154,41,128),(16,127,42,153),(17,152,43,126),(18,125,44,151),(19,150,45,124),(20,123,46,149),(21,148,47,122),(22,121,48,147),(23,146,49,120),(24,119,50,145),(25,144,51,118),(26,117,52,143),(53,188,79,162),(54,161,80,187),(55,186,81,160),(56,159,82,185),(57,184,83,158),(58,157,84,183),(59,182,85,208),(60,207,86,181),(61,180,87,206),(62,205,88,179),(63,178,89,204),(64,203,90,177),(65,176,91,202),(66,201,92,175),(67,174,93,200),(68,199,94,173),(69,172,95,198),(70,197,96,171),(71,170,97,196),(72,195,98,169),(73,168,99,194),(74,193,100,167),(75,166,101,192),(76,191,102,165),(77,164,103,190),(78,189,104,163)])`

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`

׿
×
𝔽