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## G = C25⋊C8order 200 = 23·52

### The semidirect product of C25 and C8 acting via C8/C2=C4

Aliases: C25⋊C8, C50.C4, C10.1F5, Dic25.2C2, C5.(C5⋊C8), C2.(C25⋊C4), SmallGroup(200,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C25 — C25⋊C8
 Chief series C1 — C5 — C25 — C50 — Dic25 — C25⋊C8
 Lower central C25 — C25⋊C8
 Upper central C1 — C2

Generators and relations for C25⋊C8
G = < a,b | a25=b8=1, bab-1=a18 >

Character table of C25⋊C8

 class 1 2 4A 4B 5 8A 8B 8C 8D 10 25A 25B 25C 25D 25E 50A 50B 50C 50D 50E size 1 1 25 25 4 25 25 25 25 4 4 4 4 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 1 -1 -1 1 -i i -i i 1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ4 1 1 -1 -1 1 i -i i -i 1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ5 1 -1 -i i 1 ζ8 ζ83 ζ85 ζ87 -1 1 1 1 1 1 -1 -1 -1 -1 -1 linear of order 8 ρ6 1 -1 -i i 1 ζ85 ζ87 ζ8 ζ83 -1 1 1 1 1 1 -1 -1 -1 -1 -1 linear of order 8 ρ7 1 -1 i -i 1 ζ83 ζ8 ζ87 ζ85 -1 1 1 1 1 1 -1 -1 -1 -1 -1 linear of order 8 ρ8 1 -1 i -i 1 ζ87 ζ85 ζ83 ζ8 -1 1 1 1 1 1 -1 -1 -1 -1 -1 linear of order 8 ρ9 4 4 0 0 -1 0 0 0 0 -1 ζ2522+ζ2521+ζ254+ζ253 ζ2519+ζ2517+ζ258+ζ256 ζ2523+ζ2514+ζ2511+ζ252 ζ2516+ζ2513+ζ2512+ζ259 ζ2524+ζ2518+ζ257+ζ25 ζ2522+ζ2521+ζ254+ζ253 ζ2524+ζ2518+ζ257+ζ25 ζ2519+ζ2517+ζ258+ζ256 ζ2523+ζ2514+ζ2511+ζ252 ζ2516+ζ2513+ζ2512+ζ259 orthogonal lifted from C25⋊C4 ρ10 4 4 0 0 4 0 0 0 0 4 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ11 4 4 0 0 -1 0 0 0 0 -1 ζ2516+ζ2513+ζ2512+ζ259 ζ2524+ζ2518+ζ257+ζ25 ζ2519+ζ2517+ζ258+ζ256 ζ2523+ζ2514+ζ2511+ζ252 ζ2522+ζ2521+ζ254+ζ253 ζ2516+ζ2513+ζ2512+ζ259 ζ2522+ζ2521+ζ254+ζ253 ζ2524+ζ2518+ζ257+ζ25 ζ2519+ζ2517+ζ258+ζ256 ζ2523+ζ2514+ζ2511+ζ252 orthogonal lifted from C25⋊C4 ρ12 4 4 0 0 -1 0 0 0 0 -1 ζ2519+ζ2517+ζ258+ζ256 ζ2516+ζ2513+ζ2512+ζ259 ζ2522+ζ2521+ζ254+ζ253 ζ2524+ζ2518+ζ257+ζ25 ζ2523+ζ2514+ζ2511+ζ252 ζ2519+ζ2517+ζ258+ζ256 ζ2523+ζ2514+ζ2511+ζ252 ζ2516+ζ2513+ζ2512+ζ259 ζ2522+ζ2521+ζ254+ζ253 ζ2524+ζ2518+ζ257+ζ25 orthogonal lifted from C25⋊C4 ρ13 4 4 0 0 -1 0 0 0 0 -1 ζ2524+ζ2518+ζ257+ζ25 ζ2523+ζ2514+ζ2511+ζ252 ζ2516+ζ2513+ζ2512+ζ259 ζ2522+ζ2521+ζ254+ζ253 ζ2519+ζ2517+ζ258+ζ256 ζ2524+ζ2518+ζ257+ζ25 ζ2519+ζ2517+ζ258+ζ256 ζ2523+ζ2514+ζ2511+ζ252 ζ2516+ζ2513+ζ2512+ζ259 ζ2522+ζ2521+ζ254+ζ253 orthogonal lifted from C25⋊C4 ρ14 4 4 0 0 -1 0 0 0 0 -1 ζ2523+ζ2514+ζ2511+ζ252 ζ2522+ζ2521+ζ254+ζ253 ζ2524+ζ2518+ζ257+ζ25 ζ2519+ζ2517+ζ258+ζ256 ζ2516+ζ2513+ζ2512+ζ259 ζ2523+ζ2514+ζ2511+ζ252 ζ2516+ζ2513+ζ2512+ζ259 ζ2522+ζ2521+ζ254+ζ253 ζ2524+ζ2518+ζ257+ζ25 ζ2519+ζ2517+ζ258+ζ256 orthogonal lifted from C25⋊C4 ρ15 4 -4 0 0 4 0 0 0 0 -4 -1 -1 -1 -1 -1 1 1 1 1 1 symplectic lifted from C5⋊C8, Schur index 2 ρ16 4 -4 0 0 -1 0 0 0 0 1 ζ2523+ζ2514+ζ2511+ζ252 ζ2522+ζ2521+ζ254+ζ253 ζ2524+ζ2518+ζ257+ζ25 ζ2519+ζ2517+ζ258+ζ256 ζ2516+ζ2513+ζ2512+ζ259 -ζ2523-ζ2514-ζ2511-ζ252 -ζ2516-ζ2513-ζ2512-ζ259 -ζ2522-ζ2521-ζ254-ζ253 -ζ2524-ζ2518-ζ257-ζ25 -ζ2519-ζ2517-ζ258-ζ256 symplectic faithful, Schur index 2 ρ17 4 -4 0 0 -1 0 0 0 0 1 ζ2516+ζ2513+ζ2512+ζ259 ζ2524+ζ2518+ζ257+ζ25 ζ2519+ζ2517+ζ258+ζ256 ζ2523+ζ2514+ζ2511+ζ252 ζ2522+ζ2521+ζ254+ζ253 -ζ2516-ζ2513-ζ2512-ζ259 -ζ2522-ζ2521-ζ254-ζ253 -ζ2524-ζ2518-ζ257-ζ25 -ζ2519-ζ2517-ζ258-ζ256 -ζ2523-ζ2514-ζ2511-ζ252 symplectic faithful, Schur index 2 ρ18 4 -4 0 0 -1 0 0 0 0 1 ζ2524+ζ2518+ζ257+ζ25 ζ2523+ζ2514+ζ2511+ζ252 ζ2516+ζ2513+ζ2512+ζ259 ζ2522+ζ2521+ζ254+ζ253 ζ2519+ζ2517+ζ258+ζ256 -ζ2524-ζ2518-ζ257-ζ25 -ζ2519-ζ2517-ζ258-ζ256 -ζ2523-ζ2514-ζ2511-ζ252 -ζ2516-ζ2513-ζ2512-ζ259 -ζ2522-ζ2521-ζ254-ζ253 symplectic faithful, Schur index 2 ρ19 4 -4 0 0 -1 0 0 0 0 1 ζ2519+ζ2517+ζ258+ζ256 ζ2516+ζ2513+ζ2512+ζ259 ζ2522+ζ2521+ζ254+ζ253 ζ2524+ζ2518+ζ257+ζ25 ζ2523+ζ2514+ζ2511+ζ252 -ζ2519-ζ2517-ζ258-ζ256 -ζ2523-ζ2514-ζ2511-ζ252 -ζ2516-ζ2513-ζ2512-ζ259 -ζ2522-ζ2521-ζ254-ζ253 -ζ2524-ζ2518-ζ257-ζ25 symplectic faithful, Schur index 2 ρ20 4 -4 0 0 -1 0 0 0 0 1 ζ2522+ζ2521+ζ254+ζ253 ζ2519+ζ2517+ζ258+ζ256 ζ2523+ζ2514+ζ2511+ζ252 ζ2516+ζ2513+ζ2512+ζ259 ζ2524+ζ2518+ζ257+ζ25 -ζ2522-ζ2521-ζ254-ζ253 -ζ2524-ζ2518-ζ257-ζ25 -ζ2519-ζ2517-ζ258-ζ256 -ζ2523-ζ2514-ζ2511-ζ252 -ζ2516-ζ2513-ζ2512-ζ259 symplectic faithful, Schur index 2

Smallest permutation representation of C25⋊C8
Regular action on 200 points
Generators in S200
```(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)
(1 188 83 132 38 170 62 106)(2 195 82 150 39 152 61 124)(3 177 81 143 40 159 60 117)(4 184 80 136 41 166 59 110)(5 191 79 129 42 173 58 103)(6 198 78 147 43 155 57 121)(7 180 77 140 44 162 56 114)(8 187 76 133 45 169 55 107)(9 194 100 126 46 151 54 125)(10 176 99 144 47 158 53 118)(11 183 98 137 48 165 52 111)(12 190 97 130 49 172 51 104)(13 197 96 148 50 154 75 122)(14 179 95 141 26 161 74 115)(15 186 94 134 27 168 73 108)(16 193 93 127 28 175 72 101)(17 200 92 145 29 157 71 119)(18 182 91 138 30 164 70 112)(19 189 90 131 31 171 69 105)(20 196 89 149 32 153 68 123)(21 178 88 142 33 160 67 116)(22 185 87 135 34 167 66 109)(23 192 86 128 35 174 65 102)(24 199 85 146 36 156 64 120)(25 181 84 139 37 163 63 113)```

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

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

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

C25⋊C8 is a maximal subgroup of   D25⋊C8  C100.C4  C25⋊M4(2)
C25⋊C8 is a maximal quotient of   C25⋊C16

Matrix representation of C25⋊C8 in GL4(𝔽7) generated by

 5 1 1 5 3 5 0 0 0 4 0 3 5 0 2 2
,
 1 1 5 3 4 5 2 6 2 1 6 1 1 2 6 2
`G:=sub<GL(4,GF(7))| [5,3,0,5,1,5,4,0,1,0,0,2,5,0,3,2],[1,4,2,1,1,5,1,2,5,2,6,6,3,6,1,2] >;`

C25⋊C8 in GAP, Magma, Sage, TeX

`C_{25}\rtimes C_8`
`% in TeX`

`G:=Group("C25:C8");`
`// GroupNames label`

`G:=SmallGroup(200,3);`
`// by ID`

`G=gap.SmallGroup(200,3);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-5,-5,10,26,1123,1928,118,2004,2009]);`
`// Polycyclic`

`G:=Group<a,b|a^25=b^8=1,b*a*b^-1=a^18>;`
`// generators/relations`

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