direct product, abelian, monomial, 5-elementary
Aliases: C5×C45, SmallGroup(225,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C5×C45 |
| C1 — C5×C45 |
| C1 — C5×C45 |
Generators and relations for C5×C45
G = < a,b | a5=b45=1, ab=ba >
Smallest permutation representation of C5×C45
►Regular action on 225 points
Generators in S225
(1 117 59 199 152)(2 118 60 200 153)(3 119 61 201 154)(4 120 62 202 155)(5 121 63 203 156)(6 122 64 204 157)(7 123 65 205 158)(8 124 66 206 159)(9 125 67 207 160)(10 126 68 208 161)(11 127 69 209 162)(12 128 70 210 163)(13 129 71 211 164)(14 130 72 212 165)(15 131 73 213 166)(16 132 74 214 167)(17 133 75 215 168)(18 134 76 216 169)(19 135 77 217 170)(20 91 78 218 171)(21 92 79 219 172)(22 93 80 220 173)(23 94 81 221 174)(24 95 82 222 175)(25 96 83 223 176)(26 97 84 224 177)(27 98 85 225 178)(28 99 86 181 179)(29 100 87 182 180)(30 101 88 183 136)(31 102 89 184 137)(32 103 90 185 138)(33 104 46 186 139)(34 105 47 187 140)(35 106 48 188 141)(36 107 49 189 142)(37 108 50 190 143)(38 109 51 191 144)(39 110 52 192 145)(40 111 53 193 146)(41 112 54 194 147)(42 113 55 195 148)(43 114 56 196 149)(44 115 57 197 150)(45 116 58 198 151)
(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)
G:=sub<Sym(225)| (1,117,59,199,152)(2,118,60,200,153)(3,119,61,201,154)(4,120,62,202,155)(5,121,63,203,156)(6,122,64,204,157)(7,123,65,205,158)(8,124,66,206,159)(9,125,67,207,160)(10,126,68,208,161)(11,127,69,209,162)(12,128,70,210,163)(13,129,71,211,164)(14,130,72,212,165)(15,131,73,213,166)(16,132,74,214,167)(17,133,75,215,168)(18,134,76,216,169)(19,135,77,217,170)(20,91,78,218,171)(21,92,79,219,172)(22,93,80,220,173)(23,94,81,221,174)(24,95,82,222,175)(25,96,83,223,176)(26,97,84,224,177)(27,98,85,225,178)(28,99,86,181,179)(29,100,87,182,180)(30,101,88,183,136)(31,102,89,184,137)(32,103,90,185,138)(33,104,46,186,139)(34,105,47,187,140)(35,106,48,188,141)(36,107,49,189,142)(37,108,50,190,143)(38,109,51,191,144)(39,110,52,192,145)(40,111,53,193,146)(41,112,54,194,147)(42,113,55,195,148)(43,114,56,196,149)(44,115,57,197,150)(45,116,58,198,151), (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)>;
G:=Group( (1,117,59,199,152)(2,118,60,200,153)(3,119,61,201,154)(4,120,62,202,155)(5,121,63,203,156)(6,122,64,204,157)(7,123,65,205,158)(8,124,66,206,159)(9,125,67,207,160)(10,126,68,208,161)(11,127,69,209,162)(12,128,70,210,163)(13,129,71,211,164)(14,130,72,212,165)(15,131,73,213,166)(16,132,74,214,167)(17,133,75,215,168)(18,134,76,216,169)(19,135,77,217,170)(20,91,78,218,171)(21,92,79,219,172)(22,93,80,220,173)(23,94,81,221,174)(24,95,82,222,175)(25,96,83,223,176)(26,97,84,224,177)(27,98,85,225,178)(28,99,86,181,179)(29,100,87,182,180)(30,101,88,183,136)(31,102,89,184,137)(32,103,90,185,138)(33,104,46,186,139)(34,105,47,187,140)(35,106,48,188,141)(36,107,49,189,142)(37,108,50,190,143)(38,109,51,191,144)(39,110,52,192,145)(40,111,53,193,146)(41,112,54,194,147)(42,113,55,195,148)(43,114,56,196,149)(44,115,57,197,150)(45,116,58,198,151), (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) );
G=PermutationGroup([[(1,117,59,199,152),(2,118,60,200,153),(3,119,61,201,154),(4,120,62,202,155),(5,121,63,203,156),(6,122,64,204,157),(7,123,65,205,158),(8,124,66,206,159),(9,125,67,207,160),(10,126,68,208,161),(11,127,69,209,162),(12,128,70,210,163),(13,129,71,211,164),(14,130,72,212,165),(15,131,73,213,166),(16,132,74,214,167),(17,133,75,215,168),(18,134,76,216,169),(19,135,77,217,170),(20,91,78,218,171),(21,92,79,219,172),(22,93,80,220,173),(23,94,81,221,174),(24,95,82,222,175),(25,96,83,223,176),(26,97,84,224,177),(27,98,85,225,178),(28,99,86,181,179),(29,100,87,182,180),(30,101,88,183,136),(31,102,89,184,137),(32,103,90,185,138),(33,104,46,186,139),(34,105,47,187,140),(35,106,48,188,141),(36,107,49,189,142),(37,108,50,190,143),(38,109,51,191,144),(39,110,52,192,145),(40,111,53,193,146),(41,112,54,194,147),(42,113,55,195,148),(43,114,56,196,149),(44,115,57,197,150),(45,116,58,198,151)], [(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)]])
C5×C45 is a maximal subgroup of
C5⋊D45
225 conjugacy classes
| class | 1 | 3A | 3B | 5A | ··· | 5X | 9A | ··· | 9F | 15A | ··· | 15AV | 45A | ··· | 45EN |
| order | 1 | 3 | 3 | 5 | ··· | 5 | 9 | ··· | 9 | 15 | ··· | 15 | 45 | ··· | 45 |
| size | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
225 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | |||||
| image | C1 | C3 | C5 | C9 | C15 | C45 |
| kernel | C5×C45 | C5×C15 | C45 | C52 | C15 | C5 |
| # reps | 1 | 2 | 24 | 6 | 48 | 144 |
Matrix representation of C5×C45 ►in GL2(𝔽181) generated by
| 42 | 0 |
| 0 | 125 |
| 81 | 0 |
| 0 | 82 |
G:=sub<GL(2,GF(181))| [42,0,0,125],[81,0,0,82] >;
C5×C45 in GAP, Magma, Sage, TeX
C_5\times C_{45} % in TeX
G:=Group("C5xC45"); // GroupNames label
G:=SmallGroup(225,4);
// by ID
G=gap.SmallGroup(225,4);
# by ID
G:=PCGroup([4,-3,-5,-5,-3,300]);
// Polycyclic
G:=Group<a,b|a^5=b^45=1,a*b=b*a>;
// generators/relations
Export