metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C35⋊1C8, C14.F5, C70.1C4, C10.Dic7, Dic5.2D7, C7⋊(C5⋊C8), C5⋊(C7⋊C8), C2.(C7⋊F5), (C7×Dic5).2C2, SmallGroup(280,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C35 — C35⋊C8 |
Generators and relations for C35⋊C8
G = < a,b | a35=b8=1, bab-1=a27 >
Smallest permutation representation of C35⋊C8
►Regular action on 280 points
Generators in S280
(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)(246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280)
(1 246 108 177 57 211 95 168)(2 259 137 204 58 224 89 160)(3 272 131 196 59 237 83 152)(4 250 125 188 60 215 77 144)(5 263 119 180 61 228 71 171)(6 276 113 207 62 241 100 163)(7 254 107 199 63 219 94 155)(8 267 136 191 64 232 88 147)(9 280 130 183 65 245 82 174)(10 258 124 210 66 223 76 166)(11 271 118 202 67 236 105 158)(12 249 112 194 68 214 99 150)(13 262 106 186 69 227 93 142)(14 275 135 178 70 240 87 169)(15 253 129 205 36 218 81 161)(16 266 123 197 37 231 75 153)(17 279 117 189 38 244 104 145)(18 257 111 181 39 222 98 172)(19 270 140 208 40 235 92 164)(20 248 134 200 41 213 86 156)(21 261 128 192 42 226 80 148)(22 274 122 184 43 239 74 175)(23 252 116 176 44 217 103 167)(24 265 110 203 45 230 97 159)(25 278 139 195 46 243 91 151)(26 256 133 187 47 221 85 143)(27 269 127 179 48 234 79 170)(28 247 121 206 49 212 73 162)(29 260 115 198 50 225 102 154)(30 273 109 190 51 238 96 146)(31 251 138 182 52 216 90 173)(32 264 132 209 53 229 84 165)(33 277 126 201 54 242 78 157)(34 255 120 193 55 220 72 149)(35 268 114 185 56 233 101 141)
G:=sub<Sym(280)| (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)(246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280), (1,246,108,177,57,211,95,168)(2,259,137,204,58,224,89,160)(3,272,131,196,59,237,83,152)(4,250,125,188,60,215,77,144)(5,263,119,180,61,228,71,171)(6,276,113,207,62,241,100,163)(7,254,107,199,63,219,94,155)(8,267,136,191,64,232,88,147)(9,280,130,183,65,245,82,174)(10,258,124,210,66,223,76,166)(11,271,118,202,67,236,105,158)(12,249,112,194,68,214,99,150)(13,262,106,186,69,227,93,142)(14,275,135,178,70,240,87,169)(15,253,129,205,36,218,81,161)(16,266,123,197,37,231,75,153)(17,279,117,189,38,244,104,145)(18,257,111,181,39,222,98,172)(19,270,140,208,40,235,92,164)(20,248,134,200,41,213,86,156)(21,261,128,192,42,226,80,148)(22,274,122,184,43,239,74,175)(23,252,116,176,44,217,103,167)(24,265,110,203,45,230,97,159)(25,278,139,195,46,243,91,151)(26,256,133,187,47,221,85,143)(27,269,127,179,48,234,79,170)(28,247,121,206,49,212,73,162)(29,260,115,198,50,225,102,154)(30,273,109,190,51,238,96,146)(31,251,138,182,52,216,90,173)(32,264,132,209,53,229,84,165)(33,277,126,201,54,242,78,157)(34,255,120,193,55,220,72,149)(35,268,114,185,56,233,101,141)>;
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)(246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280), (1,246,108,177,57,211,95,168)(2,259,137,204,58,224,89,160)(3,272,131,196,59,237,83,152)(4,250,125,188,60,215,77,144)(5,263,119,180,61,228,71,171)(6,276,113,207,62,241,100,163)(7,254,107,199,63,219,94,155)(8,267,136,191,64,232,88,147)(9,280,130,183,65,245,82,174)(10,258,124,210,66,223,76,166)(11,271,118,202,67,236,105,158)(12,249,112,194,68,214,99,150)(13,262,106,186,69,227,93,142)(14,275,135,178,70,240,87,169)(15,253,129,205,36,218,81,161)(16,266,123,197,37,231,75,153)(17,279,117,189,38,244,104,145)(18,257,111,181,39,222,98,172)(19,270,140,208,40,235,92,164)(20,248,134,200,41,213,86,156)(21,261,128,192,42,226,80,148)(22,274,122,184,43,239,74,175)(23,252,116,176,44,217,103,167)(24,265,110,203,45,230,97,159)(25,278,139,195,46,243,91,151)(26,256,133,187,47,221,85,143)(27,269,127,179,48,234,79,170)(28,247,121,206,49,212,73,162)(29,260,115,198,50,225,102,154)(30,273,109,190,51,238,96,146)(31,251,138,182,52,216,90,173)(32,264,132,209,53,229,84,165)(33,277,126,201,54,242,78,157)(34,255,120,193,55,220,72,149)(35,268,114,185,56,233,101,141) );
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),(246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280)], [(1,246,108,177,57,211,95,168),(2,259,137,204,58,224,89,160),(3,272,131,196,59,237,83,152),(4,250,125,188,60,215,77,144),(5,263,119,180,61,228,71,171),(6,276,113,207,62,241,100,163),(7,254,107,199,63,219,94,155),(8,267,136,191,64,232,88,147),(9,280,130,183,65,245,82,174),(10,258,124,210,66,223,76,166),(11,271,118,202,67,236,105,158),(12,249,112,194,68,214,99,150),(13,262,106,186,69,227,93,142),(14,275,135,178,70,240,87,169),(15,253,129,205,36,218,81,161),(16,266,123,197,37,231,75,153),(17,279,117,189,38,244,104,145),(18,257,111,181,39,222,98,172),(19,270,140,208,40,235,92,164),(20,248,134,200,41,213,86,156),(21,261,128,192,42,226,80,148),(22,274,122,184,43,239,74,175),(23,252,116,176,44,217,103,167),(24,265,110,203,45,230,97,159),(25,278,139,195,46,243,91,151),(26,256,133,187,47,221,85,143),(27,269,127,179,48,234,79,170),(28,247,121,206,49,212,73,162),(29,260,115,198,50,225,102,154),(30,273,109,190,51,238,96,146),(31,251,138,182,52,216,90,173),(32,264,132,209,53,229,84,165),(33,277,126,201,54,242,78,157),(34,255,120,193,55,220,72,149),(35,268,114,185,56,233,101,141)]])
34 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5 | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 10 | 14A | 14B | 14C | 28A | ··· | 28F | 35A | ··· | 35F | 70A | ··· | 70F |
| order | 1 | 2 | 4 | 4 | 5 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 10 | 14 | 14 | 14 | 28 | ··· | 28 | 35 | ··· | 35 | 70 | ··· | 70 |
| size | 1 | 1 | 5 | 5 | 4 | 2 | 2 | 2 | 35 | 35 | 35 | 35 | 4 | 2 | 2 | 2 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | - | + | - | |||||
| image | C1 | C2 | C4 | C8 | D7 | Dic7 | C7⋊C8 | F5 | C5⋊C8 | C7⋊F5 | C35⋊C8 |
| kernel | C35⋊C8 | C7×Dic5 | C70 | C35 | Dic5 | C10 | C5 | C14 | C7 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 3 | 3 | 6 | 1 | 1 | 6 | 6 |
Matrix representation of C35⋊C8 ►in GL6(𝔽281)
| 41 | 275 | 0 | 0 | 0 | 0 |
| 6 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 41 | 275 |
| 0 | 0 | 0 | 0 | 6 | 6 |
| 0 | 0 | 240 | 6 | 121 | 90 |
| 0 | 0 | 275 | 275 | 191 | 84 |
| 12 | 181 | 0 | 0 | 0 | 0 |
| 251 | 269 | 0 | 0 | 0 | 0 |
| 0 | 0 | 198 | 253 | 74 | 160 |
| 0 | 0 | 3 | 83 | 217 | 207 |
| 0 | 0 | 214 | 181 | 156 | 227 |
| 0 | 0 | 118 | 67 | 13 | 125 |
G:=sub<GL(6,GF(281))| [41,6,0,0,0,0,275,6,0,0,0,0,0,0,0,0,240,275,0,0,0,0,6,275,0,0,41,6,121,191,0,0,275,6,90,84],[12,251,0,0,0,0,181,269,0,0,0,0,0,0,198,3,214,118,0,0,253,83,181,67,0,0,74,217,156,13,0,0,160,207,227,125] >;
C35⋊C8 in GAP, Magma, Sage, TeX
C_{35}\rtimes C_8 % in TeX
G:=Group("C35:C8"); // GroupNames label
G:=SmallGroup(280,6);
// by ID
G=gap.SmallGroup(280,6);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-7,10,26,483,328,6004]);
// Polycyclic
G:=Group<a,b|a^35=b^8=1,b*a*b^-1=a^27>;
// generators/relations
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