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G = C252C8order 200 = 23·52

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

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

Aliases: C252C8, C50.2C4, C4.2D25, C20.4D5, C2.Dic25, C100.2C2, C10.2Dic5, C5.(C52C8), SmallGroup(200,1)

Series: Derived Chief Lower central Upper central

C1C25 — C252C8
C1C5C25C50C100 — C252C8
C25 — C252C8
C1C4

Generators and relations for C252C8
 G = < a,b | a25=b8=1, bab-1=a-1 >

25C8
5C52C8

Smallest permutation representation of C252C8
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)
(1 182 89 150 31 167 64 103)(2 181 90 149 32 166 65 102)(3 180 91 148 33 165 66 101)(4 179 92 147 34 164 67 125)(5 178 93 146 35 163 68 124)(6 177 94 145 36 162 69 123)(7 176 95 144 37 161 70 122)(8 200 96 143 38 160 71 121)(9 199 97 142 39 159 72 120)(10 198 98 141 40 158 73 119)(11 197 99 140 41 157 74 118)(12 196 100 139 42 156 75 117)(13 195 76 138 43 155 51 116)(14 194 77 137 44 154 52 115)(15 193 78 136 45 153 53 114)(16 192 79 135 46 152 54 113)(17 191 80 134 47 151 55 112)(18 190 81 133 48 175 56 111)(19 189 82 132 49 174 57 110)(20 188 83 131 50 173 58 109)(21 187 84 130 26 172 59 108)(22 186 85 129 27 171 60 107)(23 185 86 128 28 170 61 106)(24 184 87 127 29 169 62 105)(25 183 88 126 30 168 63 104)

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

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

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

C252C8 is a maximal subgroup of
C25⋊C16  C8×D25  C8⋊D25  C4.Dic25  D4.D25  D4⋊D25  C25⋊Q16  Q8⋊D25
C252C8 is a maximal quotient of
C252C16

56 conjugacy classes

class 1  2 4A4B5A5B8A8B8C8D10A10B20A20B20C20D25A···25J50A···50J100A···100T
order124455888810102020202025···2550···50100···100
size111122252525252222222···22···22···2

56 irreducible representations

dim1111222222
type+++-+-
imageC1C2C4C8D5Dic5C52C8D25Dic25C252C8
kernelC252C8C100C50C25C20C10C5C4C2C1
# reps1124224101020

Matrix representation of C252C8 in GL2(𝔽401) generated by

267162
140127
,
384192
12617
G:=sub<GL(2,GF(401))| [267,140,162,127],[384,126,192,17] >;

C252C8 in GAP, Magma, Sage, TeX

C_{25}\rtimes_2C_8
% in TeX

G:=Group("C25:2C8");
// GroupNames label

G:=SmallGroup(200,1);
// by ID

G=gap.SmallGroup(200,1);
# by ID

G:=PCGroup([5,-2,-2,-2,-5,-5,10,26,1443,418,4004]);
// Polycyclic

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

Export

Subgroup lattice of C252C8 in TeX

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