direct product, abelian, monomial, 5-elementary
Aliases: C5×C55, SmallGroup(275,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C5×C55 |
| C1 — C5×C55 |
| C1 — C5×C55 |
Generators and relations for C5×C55
G = < a,b | a5=b55=1, ab=ba >
Smallest permutation representation of C5×C55
►Regular action on 275 points
Generators in S275
(1 224 190 134 108)(2 225 191 135 109)(3 226 192 136 110)(4 227 193 137 56)(5 228 194 138 57)(6 229 195 139 58)(7 230 196 140 59)(8 231 197 141 60)(9 232 198 142 61)(10 233 199 143 62)(11 234 200 144 63)(12 235 201 145 64)(13 236 202 146 65)(14 237 203 147 66)(15 238 204 148 67)(16 239 205 149 68)(17 240 206 150 69)(18 241 207 151 70)(19 242 208 152 71)(20 243 209 153 72)(21 244 210 154 73)(22 245 211 155 74)(23 246 212 156 75)(24 247 213 157 76)(25 248 214 158 77)(26 249 215 159 78)(27 250 216 160 79)(28 251 217 161 80)(29 252 218 162 81)(30 253 219 163 82)(31 254 220 164 83)(32 255 166 165 84)(33 256 167 111 85)(34 257 168 112 86)(35 258 169 113 87)(36 259 170 114 88)(37 260 171 115 89)(38 261 172 116 90)(39 262 173 117 91)(40 263 174 118 92)(41 264 175 119 93)(42 265 176 120 94)(43 266 177 121 95)(44 267 178 122 96)(45 268 179 123 97)(46 269 180 124 98)(47 270 181 125 99)(48 271 182 126 100)(49 272 183 127 101)(50 273 184 128 102)(51 274 185 129 103)(52 275 186 130 104)(53 221 187 131 105)(54 222 188 132 106)(55 223 189 133 107)
(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)(221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275)
G:=sub<Sym(275)| (1,224,190,134,108)(2,225,191,135,109)(3,226,192,136,110)(4,227,193,137,56)(5,228,194,138,57)(6,229,195,139,58)(7,230,196,140,59)(8,231,197,141,60)(9,232,198,142,61)(10,233,199,143,62)(11,234,200,144,63)(12,235,201,145,64)(13,236,202,146,65)(14,237,203,147,66)(15,238,204,148,67)(16,239,205,149,68)(17,240,206,150,69)(18,241,207,151,70)(19,242,208,152,71)(20,243,209,153,72)(21,244,210,154,73)(22,245,211,155,74)(23,246,212,156,75)(24,247,213,157,76)(25,248,214,158,77)(26,249,215,159,78)(27,250,216,160,79)(28,251,217,161,80)(29,252,218,162,81)(30,253,219,163,82)(31,254,220,164,83)(32,255,166,165,84)(33,256,167,111,85)(34,257,168,112,86)(35,258,169,113,87)(36,259,170,114,88)(37,260,171,115,89)(38,261,172,116,90)(39,262,173,117,91)(40,263,174,118,92)(41,264,175,119,93)(42,265,176,120,94)(43,266,177,121,95)(44,267,178,122,96)(45,268,179,123,97)(46,269,180,124,98)(47,270,181,125,99)(48,271,182,126,100)(49,272,183,127,101)(50,273,184,128,102)(51,274,185,129,103)(52,275,186,130,104)(53,221,187,131,105)(54,222,188,132,106)(55,223,189,133,107), (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)(221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275)>;
G:=Group( (1,224,190,134,108)(2,225,191,135,109)(3,226,192,136,110)(4,227,193,137,56)(5,228,194,138,57)(6,229,195,139,58)(7,230,196,140,59)(8,231,197,141,60)(9,232,198,142,61)(10,233,199,143,62)(11,234,200,144,63)(12,235,201,145,64)(13,236,202,146,65)(14,237,203,147,66)(15,238,204,148,67)(16,239,205,149,68)(17,240,206,150,69)(18,241,207,151,70)(19,242,208,152,71)(20,243,209,153,72)(21,244,210,154,73)(22,245,211,155,74)(23,246,212,156,75)(24,247,213,157,76)(25,248,214,158,77)(26,249,215,159,78)(27,250,216,160,79)(28,251,217,161,80)(29,252,218,162,81)(30,253,219,163,82)(31,254,220,164,83)(32,255,166,165,84)(33,256,167,111,85)(34,257,168,112,86)(35,258,169,113,87)(36,259,170,114,88)(37,260,171,115,89)(38,261,172,116,90)(39,262,173,117,91)(40,263,174,118,92)(41,264,175,119,93)(42,265,176,120,94)(43,266,177,121,95)(44,267,178,122,96)(45,268,179,123,97)(46,269,180,124,98)(47,270,181,125,99)(48,271,182,126,100)(49,272,183,127,101)(50,273,184,128,102)(51,274,185,129,103)(52,275,186,130,104)(53,221,187,131,105)(54,222,188,132,106)(55,223,189,133,107), (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)(221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275) );
G=PermutationGroup([[(1,224,190,134,108),(2,225,191,135,109),(3,226,192,136,110),(4,227,193,137,56),(5,228,194,138,57),(6,229,195,139,58),(7,230,196,140,59),(8,231,197,141,60),(9,232,198,142,61),(10,233,199,143,62),(11,234,200,144,63),(12,235,201,145,64),(13,236,202,146,65),(14,237,203,147,66),(15,238,204,148,67),(16,239,205,149,68),(17,240,206,150,69),(18,241,207,151,70),(19,242,208,152,71),(20,243,209,153,72),(21,244,210,154,73),(22,245,211,155,74),(23,246,212,156,75),(24,247,213,157,76),(25,248,214,158,77),(26,249,215,159,78),(27,250,216,160,79),(28,251,217,161,80),(29,252,218,162,81),(30,253,219,163,82),(31,254,220,164,83),(32,255,166,165,84),(33,256,167,111,85),(34,257,168,112,86),(35,258,169,113,87),(36,259,170,114,88),(37,260,171,115,89),(38,261,172,116,90),(39,262,173,117,91),(40,263,174,118,92),(41,264,175,119,93),(42,265,176,120,94),(43,266,177,121,95),(44,267,178,122,96),(45,268,179,123,97),(46,269,180,124,98),(47,270,181,125,99),(48,271,182,126,100),(49,272,183,127,101),(50,273,184,128,102),(51,274,185,129,103),(52,275,186,130,104),(53,221,187,131,105),(54,222,188,132,106),(55,223,189,133,107)], [(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),(221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275)]])
275 conjugacy classes
| class | 1 | 5A | ··· | 5X | 11A | ··· | 11J | 55A | ··· | 55IF |
| order | 1 | 5 | ··· | 5 | 11 | ··· | 11 | 55 | ··· | 55 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
275 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | |||
| image | C1 | C5 | C11 | C55 |
| kernel | C5×C55 | C55 | C52 | C5 |
| # reps | 1 | 24 | 10 | 240 |
Matrix representation of C5×C55 ►in GL2(𝔽331) generated by
| 150 | 0 |
| 0 | 124 |
| 105 | 0 |
| 0 | 144 |
G:=sub<GL(2,GF(331))| [150,0,0,124],[105,0,0,144] >;
C5×C55 in GAP, Magma, Sage, TeX
C_5\times C_{55} % in TeX
G:=Group("C5xC55"); // GroupNames label
G:=SmallGroup(275,4);
// by ID
G=gap.SmallGroup(275,4);
# by ID
G:=PCGroup([3,-5,-5,-11]);
// Polycyclic
G:=Group<a,b|a^5=b^55=1,a*b=b*a>;
// generators/relations
Export