direct product, abelian, monomial, 5-elementary
Aliases: C5×C50, SmallGroup(250,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — 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
(1 231 90 144 170)(2 232 91 145 171)(3 233 92 146 172)(4 234 93 147 173)(5 235 94 148 174)(6 236 95 149 175)(7 237 96 150 176)(8 238 97 101 177)(9 239 98 102 178)(10 240 99 103 179)(11 241 100 104 180)(12 242 51 105 181)(13 243 52 106 182)(14 244 53 107 183)(15 245 54 108 184)(16 246 55 109 185)(17 247 56 110 186)(18 248 57 111 187)(19 249 58 112 188)(20 250 59 113 189)(21 201 60 114 190)(22 202 61 115 191)(23 203 62 116 192)(24 204 63 117 193)(25 205 64 118 194)(26 206 65 119 195)(27 207 66 120 196)(28 208 67 121 197)(29 209 68 122 198)(30 210 69 123 199)(31 211 70 124 200)(32 212 71 125 151)(33 213 72 126 152)(34 214 73 127 153)(35 215 74 128 154)(36 216 75 129 155)(37 217 76 130 156)(38 218 77 131 157)(39 219 78 132 158)(40 220 79 133 159)(41 221 80 134 160)(42 222 81 135 161)(43 223 82 136 162)(44 224 83 137 163)(45 225 84 138 164)(46 226 85 139 165)(47 227 86 140 166)(48 228 87 141 167)(49 229 88 142 168)(50 230 89 143 169)
(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)| (1,231,90,144,170)(2,232,91,145,171)(3,233,92,146,172)(4,234,93,147,173)(5,235,94,148,174)(6,236,95,149,175)(7,237,96,150,176)(8,238,97,101,177)(9,239,98,102,178)(10,240,99,103,179)(11,241,100,104,180)(12,242,51,105,181)(13,243,52,106,182)(14,244,53,107,183)(15,245,54,108,184)(16,246,55,109,185)(17,247,56,110,186)(18,248,57,111,187)(19,249,58,112,188)(20,250,59,113,189)(21,201,60,114,190)(22,202,61,115,191)(23,203,62,116,192)(24,204,63,117,193)(25,205,64,118,194)(26,206,65,119,195)(27,207,66,120,196)(28,208,67,121,197)(29,209,68,122,198)(30,210,69,123,199)(31,211,70,124,200)(32,212,71,125,151)(33,213,72,126,152)(34,214,73,127,153)(35,215,74,128,154)(36,216,75,129,155)(37,217,76,130,156)(38,218,77,131,157)(39,219,78,132,158)(40,220,79,133,159)(41,221,80,134,160)(42,222,81,135,161)(43,223,82,136,162)(44,224,83,137,163)(45,225,84,138,164)(46,226,85,139,165)(47,227,86,140,166)(48,228,87,141,167)(49,229,88,142,168)(50,230,89,143,169), (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( (1,231,90,144,170)(2,232,91,145,171)(3,233,92,146,172)(4,234,93,147,173)(5,235,94,148,174)(6,236,95,149,175)(7,237,96,150,176)(8,238,97,101,177)(9,239,98,102,178)(10,240,99,103,179)(11,241,100,104,180)(12,242,51,105,181)(13,243,52,106,182)(14,244,53,107,183)(15,245,54,108,184)(16,246,55,109,185)(17,247,56,110,186)(18,248,57,111,187)(19,249,58,112,188)(20,250,59,113,189)(21,201,60,114,190)(22,202,61,115,191)(23,203,62,116,192)(24,204,63,117,193)(25,205,64,118,194)(26,206,65,119,195)(27,207,66,120,196)(28,208,67,121,197)(29,209,68,122,198)(30,210,69,123,199)(31,211,70,124,200)(32,212,71,125,151)(33,213,72,126,152)(34,214,73,127,153)(35,215,74,128,154)(36,216,75,129,155)(37,217,76,130,156)(38,218,77,131,157)(39,219,78,132,158)(40,220,79,133,159)(41,221,80,134,160)(42,222,81,135,161)(43,223,82,136,162)(44,224,83,137,163)(45,225,84,138,164)(46,226,85,139,165)(47,227,86,140,166)(48,228,87,141,167)(49,229,88,142,168)(50,230,89,143,169), (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([[(1,231,90,144,170),(2,232,91,145,171),(3,233,92,146,172),(4,234,93,147,173),(5,235,94,148,174),(6,236,95,149,175),(7,237,96,150,176),(8,238,97,101,177),(9,239,98,102,178),(10,240,99,103,179),(11,241,100,104,180),(12,242,51,105,181),(13,243,52,106,182),(14,244,53,107,183),(15,245,54,108,184),(16,246,55,109,185),(17,247,56,110,186),(18,248,57,111,187),(19,249,58,112,188),(20,250,59,113,189),(21,201,60,114,190),(22,202,61,115,191),(23,203,62,116,192),(24,204,63,117,193),(25,205,64,118,194),(26,206,65,119,195),(27,207,66,120,196),(28,208,67,121,197),(29,209,68,122,198),(30,210,69,123,199),(31,211,70,124,200),(32,212,71,125,151),(33,213,72,126,152),(34,214,73,127,153),(35,215,74,128,154),(36,216,75,129,155),(37,217,76,130,156),(38,218,77,131,157),(39,219,78,132,158),(40,220,79,133,159),(41,221,80,134,160),(42,222,81,135,161),(43,223,82,136,162),(44,224,83,137,163),(45,225,84,138,164),(46,226,85,139,165),(47,227,86,140,166),(48,228,87,141,167),(49,229,88,142,168),(50,230,89,143,169)], [(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 | ··· | 5X | 10A | ··· | 10X | 25A | ··· | 25CV | 50A | ··· | 50CV |
| order | 1 | 2 | 5 | ··· | 5 | 10 | ··· | 10 | 25 | ··· | 25 | 50 | ··· | 50 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
250 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C5 | C5 | C10 | C10 | C25 | C50 |
| kernel | C5×C50 | C5×C25 | C50 | C5×C10 | C25 | C52 | C10 | C5 |
| # reps | 1 | 1 | 20 | 4 | 20 | 4 | 100 | 100 |
Matrix representation of C5×C50 ►in GL2(𝔽101) generated by
| 36 | 0 |
| 0 | 36 |
| 36 | 0 |
| 0 | 9 |
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
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