Copied to
clipboard

G = C25⋊C8order 200 = 23·52

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

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

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

C1C25 — C25⋊C8
C1C5C25C50Dic25 — C25⋊C8
C25 — C25⋊C8
C1C2

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

25C4
25C8
5Dic5
5C5⋊C8

Character table of C25⋊C8

 class 124A4B58A8B8C8D1025A25B25C25D25E50A50B50C50D50E
 size 11252542525252544444444444
ρ111111111111111111111    trivial
ρ211111-1-1-1-111111111111    linear of order 2
ρ311-1-11-ii-ii11111111111    linear of order 4
ρ411-1-11i-ii-i11111111111    linear of order 4
ρ51-1-ii1ζ8ζ83ζ85ζ87-111111-1-1-1-1-1    linear of order 8
ρ61-1-ii1ζ85ζ87ζ8ζ83-111111-1-1-1-1-1    linear of order 8
ρ71-1i-i1ζ83ζ8ζ87ζ85-111111-1-1-1-1-1    linear of order 8
ρ81-1i-i1ζ87ζ85ζ83ζ8-111111-1-1-1-1-1    linear of order 8
ρ94400-10000-1ζ25222521254253ζ25192517258256ζ252325142511252ζ251625132512259ζ2524251825725ζ25222521254253ζ2524251825725ζ25192517258256ζ252325142511252ζ251625132512259    orthogonal lifted from C25⋊C4
ρ104400400004-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from F5
ρ114400-10000-1ζ251625132512259ζ2524251825725ζ25192517258256ζ252325142511252ζ25222521254253ζ251625132512259ζ25222521254253ζ2524251825725ζ25192517258256ζ252325142511252    orthogonal lifted from C25⋊C4
ρ124400-10000-1ζ25192517258256ζ251625132512259ζ25222521254253ζ2524251825725ζ252325142511252ζ25192517258256ζ252325142511252ζ251625132512259ζ25222521254253ζ2524251825725    orthogonal lifted from C25⋊C4
ρ134400-10000-1ζ2524251825725ζ252325142511252ζ251625132512259ζ25222521254253ζ25192517258256ζ2524251825725ζ25192517258256ζ252325142511252ζ251625132512259ζ25222521254253    orthogonal lifted from C25⋊C4
ρ144400-10000-1ζ252325142511252ζ25222521254253ζ2524251825725ζ25192517258256ζ251625132512259ζ252325142511252ζ251625132512259ζ25222521254253ζ2524251825725ζ25192517258256    orthogonal lifted from C25⋊C4
ρ154-40040000-4-1-1-1-1-111111    symplectic lifted from C5⋊C8, Schur index 2
ρ164-400-100001ζ252325142511252ζ25222521254253ζ2524251825725ζ25192517258256ζ25162513251225925232514251125225162513251225925222521254253252425182572525192517258256    symplectic faithful, Schur index 2
ρ174-400-100001ζ251625132512259ζ2524251825725ζ25192517258256ζ252325142511252ζ2522252125425325162513251225925222521254253252425182572525192517258256252325142511252    symplectic faithful, Schur index 2
ρ184-400-100001ζ2524251825725ζ252325142511252ζ251625132512259ζ25222521254253ζ2519251725825625242518257252519251725825625232514251125225162513251225925222521254253    symplectic faithful, Schur index 2
ρ194-400-100001ζ25192517258256ζ251625132512259ζ25222521254253ζ2524251825725ζ25232514251125225192517258256252325142511252251625132512259252225212542532524251825725    symplectic faithful, Schur index 2
ρ204-400-100001ζ25222521254253ζ25192517258256ζ252325142511252ζ251625132512259ζ252425182572525222521254253252425182572525192517258256252325142511252251625132512259    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

5115
3500
0403
5022
,
1153
4526
2161
1262
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

Export

Subgroup lattice of C25⋊C8 in TeX
Character table of C25⋊C8 in TeX

׿
×
𝔽