metacyclic, supersoluble, monomial, Z-group, 5-hyperelementary
Aliases: C11⋊C25, C55.C5, C5.(C11⋊C5), SmallGroup(275,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — C11⋊C25 |
Generators and relations for C11⋊C25
G = < a,b | a11=b25=1, bab-1=a3 >
Smallest permutation representation of C11⋊C25
►Regular action on 275 points
Generators in S275
(1 44 186 61 262 91 227 155 150 207 109)(2 62 228 208 45 263 156 110 187 92 126)(3 209 157 93 63 46 111 127 229 264 188)(4 94 112 265 210 64 128 189 158 47 230)(5 266 129 48 95 211 190 231 113 65 159)(6 49 191 66 267 96 232 160 130 212 114)(7 67 233 213 50 268 161 115 192 97 131)(8 214 162 98 68 26 116 132 234 269 193)(9 99 117 270 215 69 133 194 163 27 235)(10 271 134 28 100 216 195 236 118 70 164)(11 29 196 71 272 76 237 165 135 217 119)(12 72 238 218 30 273 166 120 197 77 136)(13 219 167 78 73 31 121 137 239 274 198)(14 79 122 275 220 74 138 199 168 32 240)(15 251 139 33 80 221 200 241 123 75 169)(16 34 176 51 252 81 242 170 140 222 124)(17 52 243 223 35 253 171 125 177 82 141)(18 224 172 83 53 36 101 142 244 254 178)(19 84 102 255 225 54 143 179 173 37 245)(20 256 144 38 85 201 180 246 103 55 174)(21 39 181 56 257 86 247 175 145 202 104)(22 57 248 203 40 258 151 105 182 87 146)(23 204 152 88 58 41 106 147 249 259 183)(24 89 107 260 205 59 148 184 153 42 250)(25 261 149 43 90 206 185 226 108 60 154)
(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)| (1,44,186,61,262,91,227,155,150,207,109)(2,62,228,208,45,263,156,110,187,92,126)(3,209,157,93,63,46,111,127,229,264,188)(4,94,112,265,210,64,128,189,158,47,230)(5,266,129,48,95,211,190,231,113,65,159)(6,49,191,66,267,96,232,160,130,212,114)(7,67,233,213,50,268,161,115,192,97,131)(8,214,162,98,68,26,116,132,234,269,193)(9,99,117,270,215,69,133,194,163,27,235)(10,271,134,28,100,216,195,236,118,70,164)(11,29,196,71,272,76,237,165,135,217,119)(12,72,238,218,30,273,166,120,197,77,136)(13,219,167,78,73,31,121,137,239,274,198)(14,79,122,275,220,74,138,199,168,32,240)(15,251,139,33,80,221,200,241,123,75,169)(16,34,176,51,252,81,242,170,140,222,124)(17,52,243,223,35,253,171,125,177,82,141)(18,224,172,83,53,36,101,142,244,254,178)(19,84,102,255,225,54,143,179,173,37,245)(20,256,144,38,85,201,180,246,103,55,174)(21,39,181,56,257,86,247,175,145,202,104)(22,57,248,203,40,258,151,105,182,87,146)(23,204,152,88,58,41,106,147,249,259,183)(24,89,107,260,205,59,148,184,153,42,250)(25,261,149,43,90,206,185,226,108,60,154), (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( (1,44,186,61,262,91,227,155,150,207,109)(2,62,228,208,45,263,156,110,187,92,126)(3,209,157,93,63,46,111,127,229,264,188)(4,94,112,265,210,64,128,189,158,47,230)(5,266,129,48,95,211,190,231,113,65,159)(6,49,191,66,267,96,232,160,130,212,114)(7,67,233,213,50,268,161,115,192,97,131)(8,214,162,98,68,26,116,132,234,269,193)(9,99,117,270,215,69,133,194,163,27,235)(10,271,134,28,100,216,195,236,118,70,164)(11,29,196,71,272,76,237,165,135,217,119)(12,72,238,218,30,273,166,120,197,77,136)(13,219,167,78,73,31,121,137,239,274,198)(14,79,122,275,220,74,138,199,168,32,240)(15,251,139,33,80,221,200,241,123,75,169)(16,34,176,51,252,81,242,170,140,222,124)(17,52,243,223,35,253,171,125,177,82,141)(18,224,172,83,53,36,101,142,244,254,178)(19,84,102,255,225,54,143,179,173,37,245)(20,256,144,38,85,201,180,246,103,55,174)(21,39,181,56,257,86,247,175,145,202,104)(22,57,248,203,40,258,151,105,182,87,146)(23,204,152,88,58,41,106,147,249,259,183)(24,89,107,260,205,59,148,184,153,42,250)(25,261,149,43,90,206,185,226,108,60,154), (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([[(1,44,186,61,262,91,227,155,150,207,109),(2,62,228,208,45,263,156,110,187,92,126),(3,209,157,93,63,46,111,127,229,264,188),(4,94,112,265,210,64,128,189,158,47,230),(5,266,129,48,95,211,190,231,113,65,159),(6,49,191,66,267,96,232,160,130,212,114),(7,67,233,213,50,268,161,115,192,97,131),(8,214,162,98,68,26,116,132,234,269,193),(9,99,117,270,215,69,133,194,163,27,235),(10,271,134,28,100,216,195,236,118,70,164),(11,29,196,71,272,76,237,165,135,217,119),(12,72,238,218,30,273,166,120,197,77,136),(13,219,167,78,73,31,121,137,239,274,198),(14,79,122,275,220,74,138,199,168,32,240),(15,251,139,33,80,221,200,241,123,75,169),(16,34,176,51,252,81,242,170,140,222,124),(17,52,243,223,35,253,171,125,177,82,141),(18,224,172,83,53,36,101,142,244,254,178),(19,84,102,255,225,54,143,179,173,37,245),(20,256,144,38,85,201,180,246,103,55,174),(21,39,181,56,257,86,247,175,145,202,104),(22,57,248,203,40,258,151,105,182,87,146),(23,204,152,88,58,41,106,147,249,259,183),(24,89,107,260,205,59,148,184,153,42,250),(25,261,149,43,90,206,185,226,108,60,154)], [(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)]])
35 conjugacy classes
| class | 1 | 5A | 5B | 5C | 5D | 11A | 11B | 25A | ··· | 25T | 55A | ··· | 55H |
| order | 1 | 5 | 5 | 5 | 5 | 11 | 11 | 25 | ··· | 25 | 55 | ··· | 55 |
| size | 1 | 1 | 1 | 1 | 1 | 5 | 5 | 11 | ··· | 11 | 5 | ··· | 5 |
35 irreducible representations
| dim | 1 | 1 | 1 | 5 | 5 |
| type | + | ||||
| image | C1 | C5 | C25 | C11⋊C5 | C11⋊C25 |
| kernel | C11⋊C25 | C55 | C11 | C5 | C1 |
| # reps | 1 | 4 | 20 | 2 | 8 |
Matrix representation of C11⋊C25 ►in GL5(𝔽3301)
| 3300 | 1 | 0 | 0 | 0 |
| 3300 | 0 | 1 | 0 | 0 |
| 3300 | 0 | 0 | 1 | 0 |
| 3300 | 0 | 0 | 0 | 1 |
| 3201 | 101 | 3299 | 3203 | 100 |
| 3174 | 304 | 133 | 1150 | 2232 |
| 3279 | 1198 | 3194 | 2877 | 413 |
| 2603 | 2787 | 40 | 1387 | 169 |
| 1819 | 1402 | 1393 | 1109 | 1199 |
| 1627 | 2239 | 534 | 372 | 1081 |
G:=sub<GL(5,GF(3301))| [3300,3300,3300,3300,3201,1,0,0,0,101,0,1,0,0,3299,0,0,1,0,3203,0,0,0,1,100],[3174,3279,2603,1819,1627,304,1198,2787,1402,2239,133,3194,40,1393,534,1150,2877,1387,1109,372,2232,413,169,1199,1081] >;
C11⋊C25 in GAP, Magma, Sage, TeX
C_{11}\rtimes C_{25} % in TeX
G:=Group("C11:C25"); // GroupNames label
G:=SmallGroup(275,1);
// by ID
G=gap.SmallGroup(275,1);
# by ID
G:=PCGroup([3,-5,-5,-11,15,902]);
// Polycyclic
G:=Group<a,b|a^11=b^25=1,b*a*b^-1=a^3>;
// generators/relations
Export