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

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

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

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

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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