metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C31⋊C9, C93.C3, C3.(C31⋊C3), SmallGroup(279,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C31 — C31⋊C9 |
Generators and relations for C31⋊C9
G = < a,b | a31=b9=1, bab-1=a5 >
Smallest permutation representation of C31⋊C9
►Regular action on 279 points
Generators in S279
(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)(249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279)
(1 254 178 69 223 154 54 199 97)(2 279 183 70 248 128 55 193 102)(3 273 157 71 242 133 56 187 107)(4 267 162 72 236 138 57 212 112)(5 261 167 73 230 143 58 206 117)(6 255 172 74 224 148 59 200 122)(7 249 177 75 218 153 60 194 96)(8 274 182 76 243 127 61 188 101)(9 268 156 77 237 132 62 213 106)(10 262 161 78 231 137 32 207 111)(11 256 166 79 225 142 33 201 116)(12 250 171 80 219 147 34 195 121)(13 275 176 81 244 152 35 189 95)(14 269 181 82 238 126 36 214 100)(15 263 186 83 232 131 37 208 105)(16 257 160 84 226 136 38 202 110)(17 251 165 85 220 141 39 196 115)(18 276 170 86 245 146 40 190 120)(19 270 175 87 239 151 41 215 94)(20 264 180 88 233 125 42 209 99)(21 258 185 89 227 130 43 203 104)(22 252 159 90 221 135 44 197 109)(23 277 164 91 246 140 45 191 114)(24 271 169 92 240 145 46 216 119)(25 265 174 93 234 150 47 210 124)(26 259 179 63 228 155 48 204 98)(27 253 184 64 222 129 49 198 103)(28 278 158 65 247 134 50 192 108)(29 272 163 66 241 139 51 217 113)(30 266 168 67 235 144 52 211 118)(31 260 173 68 229 149 53 205 123)
G:=sub<Sym(279)| (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)(249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279), (1,254,178,69,223,154,54,199,97)(2,279,183,70,248,128,55,193,102)(3,273,157,71,242,133,56,187,107)(4,267,162,72,236,138,57,212,112)(5,261,167,73,230,143,58,206,117)(6,255,172,74,224,148,59,200,122)(7,249,177,75,218,153,60,194,96)(8,274,182,76,243,127,61,188,101)(9,268,156,77,237,132,62,213,106)(10,262,161,78,231,137,32,207,111)(11,256,166,79,225,142,33,201,116)(12,250,171,80,219,147,34,195,121)(13,275,176,81,244,152,35,189,95)(14,269,181,82,238,126,36,214,100)(15,263,186,83,232,131,37,208,105)(16,257,160,84,226,136,38,202,110)(17,251,165,85,220,141,39,196,115)(18,276,170,86,245,146,40,190,120)(19,270,175,87,239,151,41,215,94)(20,264,180,88,233,125,42,209,99)(21,258,185,89,227,130,43,203,104)(22,252,159,90,221,135,44,197,109)(23,277,164,91,246,140,45,191,114)(24,271,169,92,240,145,46,216,119)(25,265,174,93,234,150,47,210,124)(26,259,179,63,228,155,48,204,98)(27,253,184,64,222,129,49,198,103)(28,278,158,65,247,134,50,192,108)(29,272,163,66,241,139,51,217,113)(30,266,168,67,235,144,52,211,118)(31,260,173,68,229,149,53,205,123)>;
G:=Group( (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)(249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279), (1,254,178,69,223,154,54,199,97)(2,279,183,70,248,128,55,193,102)(3,273,157,71,242,133,56,187,107)(4,267,162,72,236,138,57,212,112)(5,261,167,73,230,143,58,206,117)(6,255,172,74,224,148,59,200,122)(7,249,177,75,218,153,60,194,96)(8,274,182,76,243,127,61,188,101)(9,268,156,77,237,132,62,213,106)(10,262,161,78,231,137,32,207,111)(11,256,166,79,225,142,33,201,116)(12,250,171,80,219,147,34,195,121)(13,275,176,81,244,152,35,189,95)(14,269,181,82,238,126,36,214,100)(15,263,186,83,232,131,37,208,105)(16,257,160,84,226,136,38,202,110)(17,251,165,85,220,141,39,196,115)(18,276,170,86,245,146,40,190,120)(19,270,175,87,239,151,41,215,94)(20,264,180,88,233,125,42,209,99)(21,258,185,89,227,130,43,203,104)(22,252,159,90,221,135,44,197,109)(23,277,164,91,246,140,45,191,114)(24,271,169,92,240,145,46,216,119)(25,265,174,93,234,150,47,210,124)(26,259,179,63,228,155,48,204,98)(27,253,184,64,222,129,49,198,103)(28,278,158,65,247,134,50,192,108)(29,272,163,66,241,139,51,217,113)(30,266,168,67,235,144,52,211,118)(31,260,173,68,229,149,53,205,123) );
G=PermutationGroup([[(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),(249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279)], [(1,254,178,69,223,154,54,199,97),(2,279,183,70,248,128,55,193,102),(3,273,157,71,242,133,56,187,107),(4,267,162,72,236,138,57,212,112),(5,261,167,73,230,143,58,206,117),(6,255,172,74,224,148,59,200,122),(7,249,177,75,218,153,60,194,96),(8,274,182,76,243,127,61,188,101),(9,268,156,77,237,132,62,213,106),(10,262,161,78,231,137,32,207,111),(11,256,166,79,225,142,33,201,116),(12,250,171,80,219,147,34,195,121),(13,275,176,81,244,152,35,189,95),(14,269,181,82,238,126,36,214,100),(15,263,186,83,232,131,37,208,105),(16,257,160,84,226,136,38,202,110),(17,251,165,85,220,141,39,196,115),(18,276,170,86,245,146,40,190,120),(19,270,175,87,239,151,41,215,94),(20,264,180,88,233,125,42,209,99),(21,258,185,89,227,130,43,203,104),(22,252,159,90,221,135,44,197,109),(23,277,164,91,246,140,45,191,114),(24,271,169,92,240,145,46,216,119),(25,265,174,93,234,150,47,210,124),(26,259,179,63,228,155,48,204,98),(27,253,184,64,222,129,49,198,103),(28,278,158,65,247,134,50,192,108),(29,272,163,66,241,139,51,217,113),(30,266,168,67,235,144,52,211,118),(31,260,173,68,229,149,53,205,123)]])
39 conjugacy classes
| class | 1 | 3A | 3B | 9A | ··· | 9F | 31A | ··· | 31J | 93A | ··· | 93T |
| order | 1 | 3 | 3 | 9 | ··· | 9 | 31 | ··· | 31 | 93 | ··· | 93 |
| size | 1 | 1 | 1 | 31 | ··· | 31 | 3 | ··· | 3 | 3 | ··· | 3 |
39 irreducible representations
| dim | 1 | 1 | 1 | 3 | 3 |
| type | + | ||||
| image | C1 | C3 | C9 | C31⋊C3 | C31⋊C9 |
| kernel | C31⋊C9 | C93 | C31 | C3 | C1 |
| # reps | 1 | 2 | 6 | 10 | 20 |
Matrix representation of C31⋊C9 ►in GL4(𝔽1117) generated by
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 160 | 588 |
| 777 | 0 | 0 | 0 |
| 0 | 770 | 604 | 1108 |
| 0 | 561 | 828 | 238 |
| 0 | 958 | 690 | 636 |
G:=sub<GL(4,GF(1117))| [1,0,0,0,0,0,0,1,0,1,0,160,0,0,1,588],[777,0,0,0,0,770,561,958,0,604,828,690,0,1108,238,636] >;
C31⋊C9 in GAP, Magma, Sage, TeX
C_{31}\rtimes C_9 % in TeX
G:=Group("C31:C9"); // GroupNames label
G:=SmallGroup(279,1);
// by ID
G=gap.SmallGroup(279,1);
# by ID
G:=PCGroup([3,-3,-3,-31,9,2027]);
// Polycyclic
G:=Group<a,b|a^31=b^9=1,b*a*b^-1=a^5>;
// generators/relations
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