direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C35, C4.C70, C140.7C2, C28.3C10, C20.3C14, C70.24C22, C2.2(C2×C70), C10.7(C2×C14), C14.7(C2×C10), SmallGroup(280,31)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C35
G = < a,b,c | a35=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Q8×C35
►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)
(1 219 81 163)(2 220 82 164)(3 221 83 165)(4 222 84 166)(5 223 85 167)(6 224 86 168)(7 225 87 169)(8 226 88 170)(9 227 89 171)(10 228 90 172)(11 229 91 173)(12 230 92 174)(13 231 93 175)(14 232 94 141)(15 233 95 142)(16 234 96 143)(17 235 97 144)(18 236 98 145)(19 237 99 146)(20 238 100 147)(21 239 101 148)(22 240 102 149)(23 241 103 150)(24 242 104 151)(25 243 105 152)(26 244 71 153)(27 245 72 154)(28 211 73 155)(29 212 74 156)(30 213 75 157)(31 214 76 158)(32 215 77 159)(33 216 78 160)(34 217 79 161)(35 218 80 162)(36 127 249 203)(37 128 250 204)(38 129 251 205)(39 130 252 206)(40 131 253 207)(41 132 254 208)(42 133 255 209)(43 134 256 210)(44 135 257 176)(45 136 258 177)(46 137 259 178)(47 138 260 179)(48 139 261 180)(49 140 262 181)(50 106 263 182)(51 107 264 183)(52 108 265 184)(53 109 266 185)(54 110 267 186)(55 111 268 187)(56 112 269 188)(57 113 270 189)(58 114 271 190)(59 115 272 191)(60 116 273 192)(61 117 274 193)(62 118 275 194)(63 119 276 195)(64 120 277 196)(65 121 278 197)(66 122 279 198)(67 123 280 199)(68 124 246 200)(69 125 247 201)(70 126 248 202)
(1 262 81 49)(2 263 82 50)(3 264 83 51)(4 265 84 52)(5 266 85 53)(6 267 86 54)(7 268 87 55)(8 269 88 56)(9 270 89 57)(10 271 90 58)(11 272 91 59)(12 273 92 60)(13 274 93 61)(14 275 94 62)(15 276 95 63)(16 277 96 64)(17 278 97 65)(18 279 98 66)(19 280 99 67)(20 246 100 68)(21 247 101 69)(22 248 102 70)(23 249 103 36)(24 250 104 37)(25 251 105 38)(26 252 71 39)(27 253 72 40)(28 254 73 41)(29 255 74 42)(30 256 75 43)(31 257 76 44)(32 258 77 45)(33 259 78 46)(34 260 79 47)(35 261 80 48)(106 164 182 220)(107 165 183 221)(108 166 184 222)(109 167 185 223)(110 168 186 224)(111 169 187 225)(112 170 188 226)(113 171 189 227)(114 172 190 228)(115 173 191 229)(116 174 192 230)(117 175 193 231)(118 141 194 232)(119 142 195 233)(120 143 196 234)(121 144 197 235)(122 145 198 236)(123 146 199 237)(124 147 200 238)(125 148 201 239)(126 149 202 240)(127 150 203 241)(128 151 204 242)(129 152 205 243)(130 153 206 244)(131 154 207 245)(132 155 208 211)(133 156 209 212)(134 157 210 213)(135 158 176 214)(136 159 177 215)(137 160 178 216)(138 161 179 217)(139 162 180 218)(140 163 181 219)
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), (1,219,81,163)(2,220,82,164)(3,221,83,165)(4,222,84,166)(5,223,85,167)(6,224,86,168)(7,225,87,169)(8,226,88,170)(9,227,89,171)(10,228,90,172)(11,229,91,173)(12,230,92,174)(13,231,93,175)(14,232,94,141)(15,233,95,142)(16,234,96,143)(17,235,97,144)(18,236,98,145)(19,237,99,146)(20,238,100,147)(21,239,101,148)(22,240,102,149)(23,241,103,150)(24,242,104,151)(25,243,105,152)(26,244,71,153)(27,245,72,154)(28,211,73,155)(29,212,74,156)(30,213,75,157)(31,214,76,158)(32,215,77,159)(33,216,78,160)(34,217,79,161)(35,218,80,162)(36,127,249,203)(37,128,250,204)(38,129,251,205)(39,130,252,206)(40,131,253,207)(41,132,254,208)(42,133,255,209)(43,134,256,210)(44,135,257,176)(45,136,258,177)(46,137,259,178)(47,138,260,179)(48,139,261,180)(49,140,262,181)(50,106,263,182)(51,107,264,183)(52,108,265,184)(53,109,266,185)(54,110,267,186)(55,111,268,187)(56,112,269,188)(57,113,270,189)(58,114,271,190)(59,115,272,191)(60,116,273,192)(61,117,274,193)(62,118,275,194)(63,119,276,195)(64,120,277,196)(65,121,278,197)(66,122,279,198)(67,123,280,199)(68,124,246,200)(69,125,247,201)(70,126,248,202), (1,262,81,49)(2,263,82,50)(3,264,83,51)(4,265,84,52)(5,266,85,53)(6,267,86,54)(7,268,87,55)(8,269,88,56)(9,270,89,57)(10,271,90,58)(11,272,91,59)(12,273,92,60)(13,274,93,61)(14,275,94,62)(15,276,95,63)(16,277,96,64)(17,278,97,65)(18,279,98,66)(19,280,99,67)(20,246,100,68)(21,247,101,69)(22,248,102,70)(23,249,103,36)(24,250,104,37)(25,251,105,38)(26,252,71,39)(27,253,72,40)(28,254,73,41)(29,255,74,42)(30,256,75,43)(31,257,76,44)(32,258,77,45)(33,259,78,46)(34,260,79,47)(35,261,80,48)(106,164,182,220)(107,165,183,221)(108,166,184,222)(109,167,185,223)(110,168,186,224)(111,169,187,225)(112,170,188,226)(113,171,189,227)(114,172,190,228)(115,173,191,229)(116,174,192,230)(117,175,193,231)(118,141,194,232)(119,142,195,233)(120,143,196,234)(121,144,197,235)(122,145,198,236)(123,146,199,237)(124,147,200,238)(125,148,201,239)(126,149,202,240)(127,150,203,241)(128,151,204,242)(129,152,205,243)(130,153,206,244)(131,154,207,245)(132,155,208,211)(133,156,209,212)(134,157,210,213)(135,158,176,214)(136,159,177,215)(137,160,178,216)(138,161,179,217)(139,162,180,218)(140,163,181,219)>;
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), (1,219,81,163)(2,220,82,164)(3,221,83,165)(4,222,84,166)(5,223,85,167)(6,224,86,168)(7,225,87,169)(8,226,88,170)(9,227,89,171)(10,228,90,172)(11,229,91,173)(12,230,92,174)(13,231,93,175)(14,232,94,141)(15,233,95,142)(16,234,96,143)(17,235,97,144)(18,236,98,145)(19,237,99,146)(20,238,100,147)(21,239,101,148)(22,240,102,149)(23,241,103,150)(24,242,104,151)(25,243,105,152)(26,244,71,153)(27,245,72,154)(28,211,73,155)(29,212,74,156)(30,213,75,157)(31,214,76,158)(32,215,77,159)(33,216,78,160)(34,217,79,161)(35,218,80,162)(36,127,249,203)(37,128,250,204)(38,129,251,205)(39,130,252,206)(40,131,253,207)(41,132,254,208)(42,133,255,209)(43,134,256,210)(44,135,257,176)(45,136,258,177)(46,137,259,178)(47,138,260,179)(48,139,261,180)(49,140,262,181)(50,106,263,182)(51,107,264,183)(52,108,265,184)(53,109,266,185)(54,110,267,186)(55,111,268,187)(56,112,269,188)(57,113,270,189)(58,114,271,190)(59,115,272,191)(60,116,273,192)(61,117,274,193)(62,118,275,194)(63,119,276,195)(64,120,277,196)(65,121,278,197)(66,122,279,198)(67,123,280,199)(68,124,246,200)(69,125,247,201)(70,126,248,202), (1,262,81,49)(2,263,82,50)(3,264,83,51)(4,265,84,52)(5,266,85,53)(6,267,86,54)(7,268,87,55)(8,269,88,56)(9,270,89,57)(10,271,90,58)(11,272,91,59)(12,273,92,60)(13,274,93,61)(14,275,94,62)(15,276,95,63)(16,277,96,64)(17,278,97,65)(18,279,98,66)(19,280,99,67)(20,246,100,68)(21,247,101,69)(22,248,102,70)(23,249,103,36)(24,250,104,37)(25,251,105,38)(26,252,71,39)(27,253,72,40)(28,254,73,41)(29,255,74,42)(30,256,75,43)(31,257,76,44)(32,258,77,45)(33,259,78,46)(34,260,79,47)(35,261,80,48)(106,164,182,220)(107,165,183,221)(108,166,184,222)(109,167,185,223)(110,168,186,224)(111,169,187,225)(112,170,188,226)(113,171,189,227)(114,172,190,228)(115,173,191,229)(116,174,192,230)(117,175,193,231)(118,141,194,232)(119,142,195,233)(120,143,196,234)(121,144,197,235)(122,145,198,236)(123,146,199,237)(124,147,200,238)(125,148,201,239)(126,149,202,240)(127,150,203,241)(128,151,204,242)(129,152,205,243)(130,153,206,244)(131,154,207,245)(132,155,208,211)(133,156,209,212)(134,157,210,213)(135,158,176,214)(136,159,177,215)(137,160,178,216)(138,161,179,217)(139,162,180,218)(140,163,181,219) );
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)], [(1,219,81,163),(2,220,82,164),(3,221,83,165),(4,222,84,166),(5,223,85,167),(6,224,86,168),(7,225,87,169),(8,226,88,170),(9,227,89,171),(10,228,90,172),(11,229,91,173),(12,230,92,174),(13,231,93,175),(14,232,94,141),(15,233,95,142),(16,234,96,143),(17,235,97,144),(18,236,98,145),(19,237,99,146),(20,238,100,147),(21,239,101,148),(22,240,102,149),(23,241,103,150),(24,242,104,151),(25,243,105,152),(26,244,71,153),(27,245,72,154),(28,211,73,155),(29,212,74,156),(30,213,75,157),(31,214,76,158),(32,215,77,159),(33,216,78,160),(34,217,79,161),(35,218,80,162),(36,127,249,203),(37,128,250,204),(38,129,251,205),(39,130,252,206),(40,131,253,207),(41,132,254,208),(42,133,255,209),(43,134,256,210),(44,135,257,176),(45,136,258,177),(46,137,259,178),(47,138,260,179),(48,139,261,180),(49,140,262,181),(50,106,263,182),(51,107,264,183),(52,108,265,184),(53,109,266,185),(54,110,267,186),(55,111,268,187),(56,112,269,188),(57,113,270,189),(58,114,271,190),(59,115,272,191),(60,116,273,192),(61,117,274,193),(62,118,275,194),(63,119,276,195),(64,120,277,196),(65,121,278,197),(66,122,279,198),(67,123,280,199),(68,124,246,200),(69,125,247,201),(70,126,248,202)], [(1,262,81,49),(2,263,82,50),(3,264,83,51),(4,265,84,52),(5,266,85,53),(6,267,86,54),(7,268,87,55),(8,269,88,56),(9,270,89,57),(10,271,90,58),(11,272,91,59),(12,273,92,60),(13,274,93,61),(14,275,94,62),(15,276,95,63),(16,277,96,64),(17,278,97,65),(18,279,98,66),(19,280,99,67),(20,246,100,68),(21,247,101,69),(22,248,102,70),(23,249,103,36),(24,250,104,37),(25,251,105,38),(26,252,71,39),(27,253,72,40),(28,254,73,41),(29,255,74,42),(30,256,75,43),(31,257,76,44),(32,258,77,45),(33,259,78,46),(34,260,79,47),(35,261,80,48),(106,164,182,220),(107,165,183,221),(108,166,184,222),(109,167,185,223),(110,168,186,224),(111,169,187,225),(112,170,188,226),(113,171,189,227),(114,172,190,228),(115,173,191,229),(116,174,192,230),(117,175,193,231),(118,141,194,232),(119,142,195,233),(120,143,196,234),(121,144,197,235),(122,145,198,236),(123,146,199,237),(124,147,200,238),(125,148,201,239),(126,149,202,240),(127,150,203,241),(128,151,204,242),(129,152,205,243),(130,153,206,244),(131,154,207,245),(132,155,208,211),(133,156,209,212),(134,157,210,213),(135,158,176,214),(136,159,177,215),(137,160,178,216),(138,161,179,217),(139,162,180,218),(140,163,181,219)]])
175 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 7A | ··· | 7F | 10A | 10B | 10C | 10D | 14A | ··· | 14F | 20A | ··· | 20L | 28A | ··· | 28R | 35A | ··· | 35X | 70A | ··· | 70X | 140A | ··· | 140BT |
| order | 1 | 2 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 7 | ··· | 7 | 10 | 10 | 10 | 10 | 14 | ··· | 14 | 20 | ··· | 20 | 28 | ··· | 28 | 35 | ··· | 35 | 70 | ··· | 70 | 140 | ··· | 140 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
175 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | - | |||||||||
| image | C1 | C2 | C5 | C7 | C10 | C14 | C35 | C70 | Q8 | C5×Q8 | C7×Q8 | Q8×C35 |
| kernel | Q8×C35 | C140 | C7×Q8 | C5×Q8 | C28 | C20 | Q8 | C4 | C35 | C7 | C5 | C1 |
| # reps | 1 | 3 | 4 | 6 | 12 | 18 | 24 | 72 | 1 | 4 | 6 | 24 |
Matrix representation of Q8×C35 ►in GL2(𝔽281) generated by
| 200 | 0 |
| 0 | 200 |
| 253 | 279 |
| 252 | 28 |
| 142 | 248 |
| 15 | 139 |
G:=sub<GL(2,GF(281))| [200,0,0,200],[253,252,279,28],[142,15,248,139] >;
Q8×C35 in GAP, Magma, Sage, TeX
Q_8\times C_{35} % in TeX
G:=Group("Q8xC35"); // GroupNames label
G:=SmallGroup(280,31);
// by ID
G=gap.SmallGroup(280,31);
# by ID
G:=PCGroup([5,-2,-2,-5,-7,-2,700,1421,706]);
// Polycyclic
G:=Group<a,b,c|a^35=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export