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G = C5×C50order 250 = 2·53

Abelian group of type [5,50]

direct product, abelian, monomial, 5-elementary

Aliases: C5×C50, SmallGroup(250,9)

Series: Derived Chief Lower central Upper central

C1 — C5×C50
C1C5C52C5×C25 — C5×C50
C1 — C5×C50
C1 — C5×C50

Generators and relations for C5×C50
 G = < a,b | a5=b50=1, ab=ba >


Smallest permutation representation of C5×C50
Regular action on 250 points
Generators in S250
(1 133 57 234 191)(2 134 58 235 192)(3 135 59 236 193)(4 136 60 237 194)(5 137 61 238 195)(6 138 62 239 196)(7 139 63 240 197)(8 140 64 241 198)(9 141 65 242 199)(10 142 66 243 200)(11 143 67 244 151)(12 144 68 245 152)(13 145 69 246 153)(14 146 70 247 154)(15 147 71 248 155)(16 148 72 249 156)(17 149 73 250 157)(18 150 74 201 158)(19 101 75 202 159)(20 102 76 203 160)(21 103 77 204 161)(22 104 78 205 162)(23 105 79 206 163)(24 106 80 207 164)(25 107 81 208 165)(26 108 82 209 166)(27 109 83 210 167)(28 110 84 211 168)(29 111 85 212 169)(30 112 86 213 170)(31 113 87 214 171)(32 114 88 215 172)(33 115 89 216 173)(34 116 90 217 174)(35 117 91 218 175)(36 118 92 219 176)(37 119 93 220 177)(38 120 94 221 178)(39 121 95 222 179)(40 122 96 223 180)(41 123 97 224 181)(42 124 98 225 182)(43 125 99 226 183)(44 126 100 227 184)(45 127 51 228 185)(46 128 52 229 186)(47 129 53 230 187)(48 130 54 231 188)(49 131 55 232 189)(50 132 56 233 190)
(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)(201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250)

G:=sub<Sym(250)| (1,133,57,234,191)(2,134,58,235,192)(3,135,59,236,193)(4,136,60,237,194)(5,137,61,238,195)(6,138,62,239,196)(7,139,63,240,197)(8,140,64,241,198)(9,141,65,242,199)(10,142,66,243,200)(11,143,67,244,151)(12,144,68,245,152)(13,145,69,246,153)(14,146,70,247,154)(15,147,71,248,155)(16,148,72,249,156)(17,149,73,250,157)(18,150,74,201,158)(19,101,75,202,159)(20,102,76,203,160)(21,103,77,204,161)(22,104,78,205,162)(23,105,79,206,163)(24,106,80,207,164)(25,107,81,208,165)(26,108,82,209,166)(27,109,83,210,167)(28,110,84,211,168)(29,111,85,212,169)(30,112,86,213,170)(31,113,87,214,171)(32,114,88,215,172)(33,115,89,216,173)(34,116,90,217,174)(35,117,91,218,175)(36,118,92,219,176)(37,119,93,220,177)(38,120,94,221,178)(39,121,95,222,179)(40,122,96,223,180)(41,123,97,224,181)(42,124,98,225,182)(43,125,99,226,183)(44,126,100,227,184)(45,127,51,228,185)(46,128,52,229,186)(47,129,53,230,187)(48,130,54,231,188)(49,131,55,232,189)(50,132,56,233,190), (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)(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250)>;

G:=Group( (1,133,57,234,191)(2,134,58,235,192)(3,135,59,236,193)(4,136,60,237,194)(5,137,61,238,195)(6,138,62,239,196)(7,139,63,240,197)(8,140,64,241,198)(9,141,65,242,199)(10,142,66,243,200)(11,143,67,244,151)(12,144,68,245,152)(13,145,69,246,153)(14,146,70,247,154)(15,147,71,248,155)(16,148,72,249,156)(17,149,73,250,157)(18,150,74,201,158)(19,101,75,202,159)(20,102,76,203,160)(21,103,77,204,161)(22,104,78,205,162)(23,105,79,206,163)(24,106,80,207,164)(25,107,81,208,165)(26,108,82,209,166)(27,109,83,210,167)(28,110,84,211,168)(29,111,85,212,169)(30,112,86,213,170)(31,113,87,214,171)(32,114,88,215,172)(33,115,89,216,173)(34,116,90,217,174)(35,117,91,218,175)(36,118,92,219,176)(37,119,93,220,177)(38,120,94,221,178)(39,121,95,222,179)(40,122,96,223,180)(41,123,97,224,181)(42,124,98,225,182)(43,125,99,226,183)(44,126,100,227,184)(45,127,51,228,185)(46,128,52,229,186)(47,129,53,230,187)(48,130,54,231,188)(49,131,55,232,189)(50,132,56,233,190), (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)(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250) );

G=PermutationGroup([(1,133,57,234,191),(2,134,58,235,192),(3,135,59,236,193),(4,136,60,237,194),(5,137,61,238,195),(6,138,62,239,196),(7,139,63,240,197),(8,140,64,241,198),(9,141,65,242,199),(10,142,66,243,200),(11,143,67,244,151),(12,144,68,245,152),(13,145,69,246,153),(14,146,70,247,154),(15,147,71,248,155),(16,148,72,249,156),(17,149,73,250,157),(18,150,74,201,158),(19,101,75,202,159),(20,102,76,203,160),(21,103,77,204,161),(22,104,78,205,162),(23,105,79,206,163),(24,106,80,207,164),(25,107,81,208,165),(26,108,82,209,166),(27,109,83,210,167),(28,110,84,211,168),(29,111,85,212,169),(30,112,86,213,170),(31,113,87,214,171),(32,114,88,215,172),(33,115,89,216,173),(34,116,90,217,174),(35,117,91,218,175),(36,118,92,219,176),(37,119,93,220,177),(38,120,94,221,178),(39,121,95,222,179),(40,122,96,223,180),(41,123,97,224,181),(42,124,98,225,182),(43,125,99,226,183),(44,126,100,227,184),(45,127,51,228,185),(46,128,52,229,186),(47,129,53,230,187),(48,130,54,231,188),(49,131,55,232,189),(50,132,56,233,190)], [(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),(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250)])

C5×C50 is a maximal subgroup of   C50.D5

250 conjugacy classes

class 1  2 5A···5X10A···10X25A···25CV50A···50CV
order125···510···1025···2550···50
size111···11···11···11···1

250 irreducible representations

dim11111111
type++
imageC1C2C5C5C10C10C25C50
kernelC5×C50C5×C25C50C5×C10C25C52C10C5
# reps11204204100100

Matrix representation of C5×C50 in GL2(𝔽101) generated by

360
036
,
360
09
G:=sub<GL(2,GF(101))| [36,0,0,36],[36,0,0,9] >;

C5×C50 in GAP, Magma, Sage, TeX

C_5\times C_{50}
% in TeX

G:=Group("C5xC50");
// GroupNames label

G:=SmallGroup(250,9);
// by ID

G=gap.SmallGroup(250,9);
# by ID

G:=PCGroup([4,-2,-5,-5,-5,205]);
// Polycyclic

G:=Group<a,b|a^5=b^50=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C5×C50 in TeX

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