metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C35⋊Q8, C7⋊1Dic10, C5⋊1Dic14, Dic7.D5, C14.7D10, C10.7D14, C70.7C22, Dic5.1D7, Dic35.2C2, C2.7(D5×D7), (C5×Dic7).1C2, (C7×Dic5).1C2, SmallGroup(280,13)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C35⋊Q8
G = < a,b,c | a35=b4=1, c2=b2, bab-1=a29, cac-1=a6, cbc-1=b-1 >
Smallest permutation representation of C35⋊Q8
►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)
(1 121 67 97)(2 115 68 91)(3 109 69 85)(4 138 70 79)(5 132 36 73)(6 126 37 102)(7 120 38 96)(8 114 39 90)(9 108 40 84)(10 137 41 78)(11 131 42 72)(12 125 43 101)(13 119 44 95)(14 113 45 89)(15 107 46 83)(16 136 47 77)(17 130 48 71)(18 124 49 100)(19 118 50 94)(20 112 51 88)(21 106 52 82)(22 135 53 76)(23 129 54 105)(24 123 55 99)(25 117 56 93)(26 111 57 87)(27 140 58 81)(28 134 59 75)(29 128 60 104)(30 122 61 98)(31 116 62 92)(32 110 63 86)(33 139 64 80)(34 133 65 74)(35 127 66 103)(141 217 194 247)(142 211 195 276)(143 240 196 270)(144 234 197 264)(145 228 198 258)(146 222 199 252)(147 216 200 246)(148 245 201 275)(149 239 202 269)(150 233 203 263)(151 227 204 257)(152 221 205 251)(153 215 206 280)(154 244 207 274)(155 238 208 268)(156 232 209 262)(157 226 210 256)(158 220 176 250)(159 214 177 279)(160 243 178 273)(161 237 179 267)(162 231 180 261)(163 225 181 255)(164 219 182 249)(165 213 183 278)(166 242 184 272)(167 236 185 266)(168 230 186 260)(169 224 187 254)(170 218 188 248)(171 212 189 277)(172 241 190 271)(173 235 191 265)(174 229 192 259)(175 223 193 253)
(1 199 67 146)(2 205 68 152)(3 176 69 158)(4 182 70 164)(5 188 36 170)(6 194 37 141)(7 200 38 147)(8 206 39 153)(9 177 40 159)(10 183 41 165)(11 189 42 171)(12 195 43 142)(13 201 44 148)(14 207 45 154)(15 178 46 160)(16 184 47 166)(17 190 48 172)(18 196 49 143)(19 202 50 149)(20 208 51 155)(21 179 52 161)(22 185 53 167)(23 191 54 173)(24 197 55 144)(25 203 56 150)(26 209 57 156)(27 180 58 162)(28 186 59 168)(29 192 60 174)(30 198 61 145)(31 204 62 151)(32 210 63 157)(33 181 64 163)(34 187 65 169)(35 193 66 175)(71 271 130 241)(72 277 131 212)(73 248 132 218)(74 254 133 224)(75 260 134 230)(76 266 135 236)(77 272 136 242)(78 278 137 213)(79 249 138 219)(80 255 139 225)(81 261 140 231)(82 267 106 237)(83 273 107 243)(84 279 108 214)(85 250 109 220)(86 256 110 226)(87 262 111 232)(88 268 112 238)(89 274 113 244)(90 280 114 215)(91 251 115 221)(92 257 116 227)(93 263 117 233)(94 269 118 239)(95 275 119 245)(96 246 120 216)(97 252 121 222)(98 258 122 228)(99 264 123 234)(100 270 124 240)(101 276 125 211)(102 247 126 217)(103 253 127 223)(104 259 128 229)(105 265 129 235)
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), (1,121,67,97)(2,115,68,91)(3,109,69,85)(4,138,70,79)(5,132,36,73)(6,126,37,102)(7,120,38,96)(8,114,39,90)(9,108,40,84)(10,137,41,78)(11,131,42,72)(12,125,43,101)(13,119,44,95)(14,113,45,89)(15,107,46,83)(16,136,47,77)(17,130,48,71)(18,124,49,100)(19,118,50,94)(20,112,51,88)(21,106,52,82)(22,135,53,76)(23,129,54,105)(24,123,55,99)(25,117,56,93)(26,111,57,87)(27,140,58,81)(28,134,59,75)(29,128,60,104)(30,122,61,98)(31,116,62,92)(32,110,63,86)(33,139,64,80)(34,133,65,74)(35,127,66,103)(141,217,194,247)(142,211,195,276)(143,240,196,270)(144,234,197,264)(145,228,198,258)(146,222,199,252)(147,216,200,246)(148,245,201,275)(149,239,202,269)(150,233,203,263)(151,227,204,257)(152,221,205,251)(153,215,206,280)(154,244,207,274)(155,238,208,268)(156,232,209,262)(157,226,210,256)(158,220,176,250)(159,214,177,279)(160,243,178,273)(161,237,179,267)(162,231,180,261)(163,225,181,255)(164,219,182,249)(165,213,183,278)(166,242,184,272)(167,236,185,266)(168,230,186,260)(169,224,187,254)(170,218,188,248)(171,212,189,277)(172,241,190,271)(173,235,191,265)(174,229,192,259)(175,223,193,253), (1,199,67,146)(2,205,68,152)(3,176,69,158)(4,182,70,164)(5,188,36,170)(6,194,37,141)(7,200,38,147)(8,206,39,153)(9,177,40,159)(10,183,41,165)(11,189,42,171)(12,195,43,142)(13,201,44,148)(14,207,45,154)(15,178,46,160)(16,184,47,166)(17,190,48,172)(18,196,49,143)(19,202,50,149)(20,208,51,155)(21,179,52,161)(22,185,53,167)(23,191,54,173)(24,197,55,144)(25,203,56,150)(26,209,57,156)(27,180,58,162)(28,186,59,168)(29,192,60,174)(30,198,61,145)(31,204,62,151)(32,210,63,157)(33,181,64,163)(34,187,65,169)(35,193,66,175)(71,271,130,241)(72,277,131,212)(73,248,132,218)(74,254,133,224)(75,260,134,230)(76,266,135,236)(77,272,136,242)(78,278,137,213)(79,249,138,219)(80,255,139,225)(81,261,140,231)(82,267,106,237)(83,273,107,243)(84,279,108,214)(85,250,109,220)(86,256,110,226)(87,262,111,232)(88,268,112,238)(89,274,113,244)(90,280,114,215)(91,251,115,221)(92,257,116,227)(93,263,117,233)(94,269,118,239)(95,275,119,245)(96,246,120,216)(97,252,121,222)(98,258,122,228)(99,264,123,234)(100,270,124,240)(101,276,125,211)(102,247,126,217)(103,253,127,223)(104,259,128,229)(105,265,129,235)>;
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), (1,121,67,97)(2,115,68,91)(3,109,69,85)(4,138,70,79)(5,132,36,73)(6,126,37,102)(7,120,38,96)(8,114,39,90)(9,108,40,84)(10,137,41,78)(11,131,42,72)(12,125,43,101)(13,119,44,95)(14,113,45,89)(15,107,46,83)(16,136,47,77)(17,130,48,71)(18,124,49,100)(19,118,50,94)(20,112,51,88)(21,106,52,82)(22,135,53,76)(23,129,54,105)(24,123,55,99)(25,117,56,93)(26,111,57,87)(27,140,58,81)(28,134,59,75)(29,128,60,104)(30,122,61,98)(31,116,62,92)(32,110,63,86)(33,139,64,80)(34,133,65,74)(35,127,66,103)(141,217,194,247)(142,211,195,276)(143,240,196,270)(144,234,197,264)(145,228,198,258)(146,222,199,252)(147,216,200,246)(148,245,201,275)(149,239,202,269)(150,233,203,263)(151,227,204,257)(152,221,205,251)(153,215,206,280)(154,244,207,274)(155,238,208,268)(156,232,209,262)(157,226,210,256)(158,220,176,250)(159,214,177,279)(160,243,178,273)(161,237,179,267)(162,231,180,261)(163,225,181,255)(164,219,182,249)(165,213,183,278)(166,242,184,272)(167,236,185,266)(168,230,186,260)(169,224,187,254)(170,218,188,248)(171,212,189,277)(172,241,190,271)(173,235,191,265)(174,229,192,259)(175,223,193,253), (1,199,67,146)(2,205,68,152)(3,176,69,158)(4,182,70,164)(5,188,36,170)(6,194,37,141)(7,200,38,147)(8,206,39,153)(9,177,40,159)(10,183,41,165)(11,189,42,171)(12,195,43,142)(13,201,44,148)(14,207,45,154)(15,178,46,160)(16,184,47,166)(17,190,48,172)(18,196,49,143)(19,202,50,149)(20,208,51,155)(21,179,52,161)(22,185,53,167)(23,191,54,173)(24,197,55,144)(25,203,56,150)(26,209,57,156)(27,180,58,162)(28,186,59,168)(29,192,60,174)(30,198,61,145)(31,204,62,151)(32,210,63,157)(33,181,64,163)(34,187,65,169)(35,193,66,175)(71,271,130,241)(72,277,131,212)(73,248,132,218)(74,254,133,224)(75,260,134,230)(76,266,135,236)(77,272,136,242)(78,278,137,213)(79,249,138,219)(80,255,139,225)(81,261,140,231)(82,267,106,237)(83,273,107,243)(84,279,108,214)(85,250,109,220)(86,256,110,226)(87,262,111,232)(88,268,112,238)(89,274,113,244)(90,280,114,215)(91,251,115,221)(92,257,116,227)(93,263,117,233)(94,269,118,239)(95,275,119,245)(96,246,120,216)(97,252,121,222)(98,258,122,228)(99,264,123,234)(100,270,124,240)(101,276,125,211)(102,247,126,217)(103,253,127,223)(104,259,128,229)(105,265,129,235) );
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)], [(1,121,67,97),(2,115,68,91),(3,109,69,85),(4,138,70,79),(5,132,36,73),(6,126,37,102),(7,120,38,96),(8,114,39,90),(9,108,40,84),(10,137,41,78),(11,131,42,72),(12,125,43,101),(13,119,44,95),(14,113,45,89),(15,107,46,83),(16,136,47,77),(17,130,48,71),(18,124,49,100),(19,118,50,94),(20,112,51,88),(21,106,52,82),(22,135,53,76),(23,129,54,105),(24,123,55,99),(25,117,56,93),(26,111,57,87),(27,140,58,81),(28,134,59,75),(29,128,60,104),(30,122,61,98),(31,116,62,92),(32,110,63,86),(33,139,64,80),(34,133,65,74),(35,127,66,103),(141,217,194,247),(142,211,195,276),(143,240,196,270),(144,234,197,264),(145,228,198,258),(146,222,199,252),(147,216,200,246),(148,245,201,275),(149,239,202,269),(150,233,203,263),(151,227,204,257),(152,221,205,251),(153,215,206,280),(154,244,207,274),(155,238,208,268),(156,232,209,262),(157,226,210,256),(158,220,176,250),(159,214,177,279),(160,243,178,273),(161,237,179,267),(162,231,180,261),(163,225,181,255),(164,219,182,249),(165,213,183,278),(166,242,184,272),(167,236,185,266),(168,230,186,260),(169,224,187,254),(170,218,188,248),(171,212,189,277),(172,241,190,271),(173,235,191,265),(174,229,192,259),(175,223,193,253)], [(1,199,67,146),(2,205,68,152),(3,176,69,158),(4,182,70,164),(5,188,36,170),(6,194,37,141),(7,200,38,147),(8,206,39,153),(9,177,40,159),(10,183,41,165),(11,189,42,171),(12,195,43,142),(13,201,44,148),(14,207,45,154),(15,178,46,160),(16,184,47,166),(17,190,48,172),(18,196,49,143),(19,202,50,149),(20,208,51,155),(21,179,52,161),(22,185,53,167),(23,191,54,173),(24,197,55,144),(25,203,56,150),(26,209,57,156),(27,180,58,162),(28,186,59,168),(29,192,60,174),(30,198,61,145),(31,204,62,151),(32,210,63,157),(33,181,64,163),(34,187,65,169),(35,193,66,175),(71,271,130,241),(72,277,131,212),(73,248,132,218),(74,254,133,224),(75,260,134,230),(76,266,135,236),(77,272,136,242),(78,278,137,213),(79,249,138,219),(80,255,139,225),(81,261,140,231),(82,267,106,237),(83,273,107,243),(84,279,108,214),(85,250,109,220),(86,256,110,226),(87,262,111,232),(88,268,112,238),(89,274,113,244),(90,280,114,215),(91,251,115,221),(92,257,116,227),(93,263,117,233),(94,269,118,239),(95,275,119,245),(96,246,120,216),(97,252,121,222),(98,258,122,228),(99,264,123,234),(100,270,124,240),(101,276,125,211),(102,247,126,217),(103,253,127,223),(104,259,128,229),(105,265,129,235)]])
37 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 5A | 5B | 7A | 7B | 7C | 10A | 10B | 14A | 14B | 14C | 20A | 20B | 20C | 20D | 28A | ··· | 28F | 35A | ··· | 35F | 70A | ··· | 70F |
| order | 1 | 2 | 4 | 4 | 4 | 5 | 5 | 7 | 7 | 7 | 10 | 10 | 14 | 14 | 14 | 20 | 20 | 20 | 20 | 28 | ··· | 28 | 35 | ··· | 35 | 70 | ··· | 70 |
| size | 1 | 1 | 10 | 14 | 70 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
37 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | - | + | + | + | + | - | - | + | - |
| image | C1 | C2 | C2 | C2 | Q8 | D5 | D7 | D10 | D14 | Dic10 | Dic14 | D5×D7 | C35⋊Q8 |
| kernel | C35⋊Q8 | C7×Dic5 | C5×Dic7 | Dic35 | C35 | Dic7 | Dic5 | C14 | C10 | C7 | C5 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 2 | 3 | 4 | 6 | 6 | 6 |
Matrix representation of C35⋊Q8 ►in GL4(𝔽281) generated by
| 0 | 1 | 0 | 0 |
| 280 | 243 | 0 | 0 |
| 0 | 0 | 39 | 241 |
| 0 | 0 | 80 | 235 |
| 104 | 153 | 0 | 0 |
| 135 | 177 | 0 | 0 |
| 0 | 0 | 45 | 156 |
| 0 | 0 | 250 | 236 |
| 84 | 34 | 0 | 0 |
| 247 | 197 | 0 | 0 |
| 0 | 0 | 199 | 236 |
| 0 | 0 | 87 | 82 |
G:=sub<GL(4,GF(281))| [0,280,0,0,1,243,0,0,0,0,39,80,0,0,241,235],[104,135,0,0,153,177,0,0,0,0,45,250,0,0,156,236],[84,247,0,0,34,197,0,0,0,0,199,87,0,0,236,82] >;
C35⋊Q8 in GAP, Magma, Sage, TeX
C_{35}\rtimes Q_8 % in TeX
G:=Group("C35:Q8"); // GroupNames label
G:=SmallGroup(280,13);
// by ID
G=gap.SmallGroup(280,13);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-7,20,61,26,328,6004]);
// Polycyclic
G:=Group<a,b,c|a^35=b^4=1,c^2=b^2,b*a*b^-1=a^29,c*a*c^-1=a^6,c*b*c^-1=b^-1>;
// generators/relations
Export