metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C35⋊3C8, C70.3C4, C28.2D5, C4.2D35, C20.2D7, C14.Dic5, C2.Dic35, C140.2C2, C10.2Dic7, C7⋊(C5⋊2C8), C5⋊2(C7⋊C8), SmallGroup(280,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C35 — C35⋊3C8 |
Generators and relations for C35⋊3C8
G = < a,b | a35=b8=1, bab-1=a-1 >
Smallest permutation representation of C35⋊3C8
►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)
(1 259 135 189 55 224 79 173)(2 258 136 188 56 223 80 172)(3 257 137 187 57 222 81 171)(4 256 138 186 58 221 82 170)(5 255 139 185 59 220 83 169)(6 254 140 184 60 219 84 168)(7 253 106 183 61 218 85 167)(8 252 107 182 62 217 86 166)(9 251 108 181 63 216 87 165)(10 250 109 180 64 215 88 164)(11 249 110 179 65 214 89 163)(12 248 111 178 66 213 90 162)(13 247 112 177 67 212 91 161)(14 246 113 176 68 211 92 160)(15 280 114 210 69 245 93 159)(16 279 115 209 70 244 94 158)(17 278 116 208 36 243 95 157)(18 277 117 207 37 242 96 156)(19 276 118 206 38 241 97 155)(20 275 119 205 39 240 98 154)(21 274 120 204 40 239 99 153)(22 273 121 203 41 238 100 152)(23 272 122 202 42 237 101 151)(24 271 123 201 43 236 102 150)(25 270 124 200 44 235 103 149)(26 269 125 199 45 234 104 148)(27 268 126 198 46 233 105 147)(28 267 127 197 47 232 71 146)(29 266 128 196 48 231 72 145)(30 265 129 195 49 230 73 144)(31 264 130 194 50 229 74 143)(32 263 131 193 51 228 75 142)(33 262 132 192 52 227 76 141)(34 261 133 191 53 226 77 175)(35 260 134 190 54 225 78 174)
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), (1,259,135,189,55,224,79,173)(2,258,136,188,56,223,80,172)(3,257,137,187,57,222,81,171)(4,256,138,186,58,221,82,170)(5,255,139,185,59,220,83,169)(6,254,140,184,60,219,84,168)(7,253,106,183,61,218,85,167)(8,252,107,182,62,217,86,166)(9,251,108,181,63,216,87,165)(10,250,109,180,64,215,88,164)(11,249,110,179,65,214,89,163)(12,248,111,178,66,213,90,162)(13,247,112,177,67,212,91,161)(14,246,113,176,68,211,92,160)(15,280,114,210,69,245,93,159)(16,279,115,209,70,244,94,158)(17,278,116,208,36,243,95,157)(18,277,117,207,37,242,96,156)(19,276,118,206,38,241,97,155)(20,275,119,205,39,240,98,154)(21,274,120,204,40,239,99,153)(22,273,121,203,41,238,100,152)(23,272,122,202,42,237,101,151)(24,271,123,201,43,236,102,150)(25,270,124,200,44,235,103,149)(26,269,125,199,45,234,104,148)(27,268,126,198,46,233,105,147)(28,267,127,197,47,232,71,146)(29,266,128,196,48,231,72,145)(30,265,129,195,49,230,73,144)(31,264,130,194,50,229,74,143)(32,263,131,193,51,228,75,142)(33,262,132,192,52,227,76,141)(34,261,133,191,53,226,77,175)(35,260,134,190,54,225,78,174)>;
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), (1,259,135,189,55,224,79,173)(2,258,136,188,56,223,80,172)(3,257,137,187,57,222,81,171)(4,256,138,186,58,221,82,170)(5,255,139,185,59,220,83,169)(6,254,140,184,60,219,84,168)(7,253,106,183,61,218,85,167)(8,252,107,182,62,217,86,166)(9,251,108,181,63,216,87,165)(10,250,109,180,64,215,88,164)(11,249,110,179,65,214,89,163)(12,248,111,178,66,213,90,162)(13,247,112,177,67,212,91,161)(14,246,113,176,68,211,92,160)(15,280,114,210,69,245,93,159)(16,279,115,209,70,244,94,158)(17,278,116,208,36,243,95,157)(18,277,117,207,37,242,96,156)(19,276,118,206,38,241,97,155)(20,275,119,205,39,240,98,154)(21,274,120,204,40,239,99,153)(22,273,121,203,41,238,100,152)(23,272,122,202,42,237,101,151)(24,271,123,201,43,236,102,150)(25,270,124,200,44,235,103,149)(26,269,125,199,45,234,104,148)(27,268,126,198,46,233,105,147)(28,267,127,197,47,232,71,146)(29,266,128,196,48,231,72,145)(30,265,129,195,49,230,73,144)(31,264,130,194,50,229,74,143)(32,263,131,193,51,228,75,142)(33,262,132,192,52,227,76,141)(34,261,133,191,53,226,77,175)(35,260,134,190,54,225,78,174) );
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)], [(1,259,135,189,55,224,79,173),(2,258,136,188,56,223,80,172),(3,257,137,187,57,222,81,171),(4,256,138,186,58,221,82,170),(5,255,139,185,59,220,83,169),(6,254,140,184,60,219,84,168),(7,253,106,183,61,218,85,167),(8,252,107,182,62,217,86,166),(9,251,108,181,63,216,87,165),(10,250,109,180,64,215,88,164),(11,249,110,179,65,214,89,163),(12,248,111,178,66,213,90,162),(13,247,112,177,67,212,91,161),(14,246,113,176,68,211,92,160),(15,280,114,210,69,245,93,159),(16,279,115,209,70,244,94,158),(17,278,116,208,36,243,95,157),(18,277,117,207,37,242,96,156),(19,276,118,206,38,241,97,155),(20,275,119,205,39,240,98,154),(21,274,120,204,40,239,99,153),(22,273,121,203,41,238,100,152),(23,272,122,202,42,237,101,151),(24,271,123,201,43,236,102,150),(25,270,124,200,44,235,103,149),(26,269,125,199,45,234,104,148),(27,268,126,198,46,233,105,147),(28,267,127,197,47,232,71,146),(29,266,128,196,48,231,72,145),(30,265,129,195,49,230,73,144),(31,264,130,194,50,229,74,143),(32,263,131,193,51,228,75,142),(33,262,132,192,52,227,76,141),(34,261,133,191,53,226,77,175),(35,260,134,190,54,225,78,174)]])
76 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | 5B | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 10A | 10B | 14A | 14B | 14C | 20A | 20B | 20C | 20D | 28A | ··· | 28F | 35A | ··· | 35L | 70A | ··· | 70L | 140A | ··· | 140X |
| order | 1 | 2 | 4 | 4 | 5 | 5 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 10 | 10 | 14 | 14 | 14 | 20 | 20 | 20 | 20 | 28 | ··· | 28 | 35 | ··· | 35 | 70 | ··· | 70 | 140 | ··· | 140 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 35 | 35 | 35 | 35 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
76 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | - | + | - | |||||
| image | C1 | C2 | C4 | C8 | D5 | D7 | Dic5 | Dic7 | C5⋊2C8 | C7⋊C8 | D35 | Dic35 | C35⋊3C8 |
| kernel | C35⋊3C8 | C140 | C70 | C35 | C28 | C20 | C14 | C10 | C7 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 2 | 3 | 2 | 3 | 4 | 6 | 12 | 12 | 24 |
Matrix representation of C35⋊3C8 ►in GL3(𝔽281) generated by
| 1 | 0 | 0 |
| 0 | 44 | 19 |
| 0 | 262 | 94 |
| 89 | 0 | 0 |
| 0 | 206 | 12 |
| 0 | 140 | 75 |
G:=sub<GL(3,GF(281))| [1,0,0,0,44,262,0,19,94],[89,0,0,0,206,140,0,12,75] >;
C35⋊3C8 in GAP, Magma, Sage, TeX
C_{35}\rtimes_3C_8 % in TeX
G:=Group("C35:3C8"); // GroupNames label
G:=SmallGroup(280,3);
// by ID
G=gap.SmallGroup(280,3);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-7,10,26,643,6004]);
// Polycyclic
G:=Group<a,b|a^35=b^8=1,b*a*b^-1=a^-1>;
// generators/relations
Export