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G = Q8×C35order 280 = 23·5·7

Direct product of C35 and Q8

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

C1C2 — Q8×C35
C1C2C14C70C140 — Q8×C35
C1C2 — Q8×C35
C1C70 — Q8×C35

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)
(1 135 59 142)(2 136 60 143)(3 137 61 144)(4 138 62 145)(5 139 63 146)(6 140 64 147)(7 106 65 148)(8 107 66 149)(9 108 67 150)(10 109 68 151)(11 110 69 152)(12 111 70 153)(13 112 36 154)(14 113 37 155)(15 114 38 156)(16 115 39 157)(17 116 40 158)(18 117 41 159)(19 118 42 160)(20 119 43 161)(21 120 44 162)(22 121 45 163)(23 122 46 164)(24 123 47 165)(25 124 48 166)(26 125 49 167)(27 126 50 168)(28 127 51 169)(29 128 52 170)(30 129 53 171)(31 130 54 172)(32 131 55 173)(33 132 56 174)(34 133 57 175)(35 134 58 141)(71 245 178 250)(72 211 179 251)(73 212 180 252)(74 213 181 253)(75 214 182 254)(76 215 183 255)(77 216 184 256)(78 217 185 257)(79 218 186 258)(80 219 187 259)(81 220 188 260)(82 221 189 261)(83 222 190 262)(84 223 191 263)(85 224 192 264)(86 225 193 265)(87 226 194 266)(88 227 195 267)(89 228 196 268)(90 229 197 269)(91 230 198 270)(92 231 199 271)(93 232 200 272)(94 233 201 273)(95 234 202 274)(96 235 203 275)(97 236 204 276)(98 237 205 277)(99 238 206 278)(100 239 207 279)(101 240 208 280)(102 241 209 246)(103 242 210 247)(104 243 176 248)(105 244 177 249)
(1 186 59 79)(2 187 60 80)(3 188 61 81)(4 189 62 82)(5 190 63 83)(6 191 64 84)(7 192 65 85)(8 193 66 86)(9 194 67 87)(10 195 68 88)(11 196 69 89)(12 197 70 90)(13 198 36 91)(14 199 37 92)(15 200 38 93)(16 201 39 94)(17 202 40 95)(18 203 41 96)(19 204 42 97)(20 205 43 98)(21 206 44 99)(22 207 45 100)(23 208 46 101)(24 209 47 102)(25 210 48 103)(26 176 49 104)(27 177 50 105)(28 178 51 71)(29 179 52 72)(30 180 53 73)(31 181 54 74)(32 182 55 75)(33 183 56 76)(34 184 57 77)(35 185 58 78)(106 224 148 264)(107 225 149 265)(108 226 150 266)(109 227 151 267)(110 228 152 268)(111 229 153 269)(112 230 154 270)(113 231 155 271)(114 232 156 272)(115 233 157 273)(116 234 158 274)(117 235 159 275)(118 236 160 276)(119 237 161 277)(120 238 162 278)(121 239 163 279)(122 240 164 280)(123 241 165 246)(124 242 166 247)(125 243 167 248)(126 244 168 249)(127 245 169 250)(128 211 170 251)(129 212 171 252)(130 213 172 253)(131 214 173 254)(132 215 174 255)(133 216 175 256)(134 217 141 257)(135 218 142 258)(136 219 143 259)(137 220 144 260)(138 221 145 261)(139 222 146 262)(140 223 147 263)

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), (1,135,59,142)(2,136,60,143)(3,137,61,144)(4,138,62,145)(5,139,63,146)(6,140,64,147)(7,106,65,148)(8,107,66,149)(9,108,67,150)(10,109,68,151)(11,110,69,152)(12,111,70,153)(13,112,36,154)(14,113,37,155)(15,114,38,156)(16,115,39,157)(17,116,40,158)(18,117,41,159)(19,118,42,160)(20,119,43,161)(21,120,44,162)(22,121,45,163)(23,122,46,164)(24,123,47,165)(25,124,48,166)(26,125,49,167)(27,126,50,168)(28,127,51,169)(29,128,52,170)(30,129,53,171)(31,130,54,172)(32,131,55,173)(33,132,56,174)(34,133,57,175)(35,134,58,141)(71,245,178,250)(72,211,179,251)(73,212,180,252)(74,213,181,253)(75,214,182,254)(76,215,183,255)(77,216,184,256)(78,217,185,257)(79,218,186,258)(80,219,187,259)(81,220,188,260)(82,221,189,261)(83,222,190,262)(84,223,191,263)(85,224,192,264)(86,225,193,265)(87,226,194,266)(88,227,195,267)(89,228,196,268)(90,229,197,269)(91,230,198,270)(92,231,199,271)(93,232,200,272)(94,233,201,273)(95,234,202,274)(96,235,203,275)(97,236,204,276)(98,237,205,277)(99,238,206,278)(100,239,207,279)(101,240,208,280)(102,241,209,246)(103,242,210,247)(104,243,176,248)(105,244,177,249), (1,186,59,79)(2,187,60,80)(3,188,61,81)(4,189,62,82)(5,190,63,83)(6,191,64,84)(7,192,65,85)(8,193,66,86)(9,194,67,87)(10,195,68,88)(11,196,69,89)(12,197,70,90)(13,198,36,91)(14,199,37,92)(15,200,38,93)(16,201,39,94)(17,202,40,95)(18,203,41,96)(19,204,42,97)(20,205,43,98)(21,206,44,99)(22,207,45,100)(23,208,46,101)(24,209,47,102)(25,210,48,103)(26,176,49,104)(27,177,50,105)(28,178,51,71)(29,179,52,72)(30,180,53,73)(31,181,54,74)(32,182,55,75)(33,183,56,76)(34,184,57,77)(35,185,58,78)(106,224,148,264)(107,225,149,265)(108,226,150,266)(109,227,151,267)(110,228,152,268)(111,229,153,269)(112,230,154,270)(113,231,155,271)(114,232,156,272)(115,233,157,273)(116,234,158,274)(117,235,159,275)(118,236,160,276)(119,237,161,277)(120,238,162,278)(121,239,163,279)(122,240,164,280)(123,241,165,246)(124,242,166,247)(125,243,167,248)(126,244,168,249)(127,245,169,250)(128,211,170,251)(129,212,171,252)(130,213,172,253)(131,214,173,254)(132,215,174,255)(133,216,175,256)(134,217,141,257)(135,218,142,258)(136,219,143,259)(137,220,144,260)(138,221,145,261)(139,222,146,262)(140,223,147,263)>;

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), (1,135,59,142)(2,136,60,143)(3,137,61,144)(4,138,62,145)(5,139,63,146)(6,140,64,147)(7,106,65,148)(8,107,66,149)(9,108,67,150)(10,109,68,151)(11,110,69,152)(12,111,70,153)(13,112,36,154)(14,113,37,155)(15,114,38,156)(16,115,39,157)(17,116,40,158)(18,117,41,159)(19,118,42,160)(20,119,43,161)(21,120,44,162)(22,121,45,163)(23,122,46,164)(24,123,47,165)(25,124,48,166)(26,125,49,167)(27,126,50,168)(28,127,51,169)(29,128,52,170)(30,129,53,171)(31,130,54,172)(32,131,55,173)(33,132,56,174)(34,133,57,175)(35,134,58,141)(71,245,178,250)(72,211,179,251)(73,212,180,252)(74,213,181,253)(75,214,182,254)(76,215,183,255)(77,216,184,256)(78,217,185,257)(79,218,186,258)(80,219,187,259)(81,220,188,260)(82,221,189,261)(83,222,190,262)(84,223,191,263)(85,224,192,264)(86,225,193,265)(87,226,194,266)(88,227,195,267)(89,228,196,268)(90,229,197,269)(91,230,198,270)(92,231,199,271)(93,232,200,272)(94,233,201,273)(95,234,202,274)(96,235,203,275)(97,236,204,276)(98,237,205,277)(99,238,206,278)(100,239,207,279)(101,240,208,280)(102,241,209,246)(103,242,210,247)(104,243,176,248)(105,244,177,249), (1,186,59,79)(2,187,60,80)(3,188,61,81)(4,189,62,82)(5,190,63,83)(6,191,64,84)(7,192,65,85)(8,193,66,86)(9,194,67,87)(10,195,68,88)(11,196,69,89)(12,197,70,90)(13,198,36,91)(14,199,37,92)(15,200,38,93)(16,201,39,94)(17,202,40,95)(18,203,41,96)(19,204,42,97)(20,205,43,98)(21,206,44,99)(22,207,45,100)(23,208,46,101)(24,209,47,102)(25,210,48,103)(26,176,49,104)(27,177,50,105)(28,178,51,71)(29,179,52,72)(30,180,53,73)(31,181,54,74)(32,182,55,75)(33,183,56,76)(34,184,57,77)(35,185,58,78)(106,224,148,264)(107,225,149,265)(108,226,150,266)(109,227,151,267)(110,228,152,268)(111,229,153,269)(112,230,154,270)(113,231,155,271)(114,232,156,272)(115,233,157,273)(116,234,158,274)(117,235,159,275)(118,236,160,276)(119,237,161,277)(120,238,162,278)(121,239,163,279)(122,240,164,280)(123,241,165,246)(124,242,166,247)(125,243,167,248)(126,244,168,249)(127,245,169,250)(128,211,170,251)(129,212,171,252)(130,213,172,253)(131,214,173,254)(132,215,174,255)(133,216,175,256)(134,217,141,257)(135,218,142,258)(136,219,143,259)(137,220,144,260)(138,221,145,261)(139,222,146,262)(140,223,147,263) );

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)], [(1,135,59,142),(2,136,60,143),(3,137,61,144),(4,138,62,145),(5,139,63,146),(6,140,64,147),(7,106,65,148),(8,107,66,149),(9,108,67,150),(10,109,68,151),(11,110,69,152),(12,111,70,153),(13,112,36,154),(14,113,37,155),(15,114,38,156),(16,115,39,157),(17,116,40,158),(18,117,41,159),(19,118,42,160),(20,119,43,161),(21,120,44,162),(22,121,45,163),(23,122,46,164),(24,123,47,165),(25,124,48,166),(26,125,49,167),(27,126,50,168),(28,127,51,169),(29,128,52,170),(30,129,53,171),(31,130,54,172),(32,131,55,173),(33,132,56,174),(34,133,57,175),(35,134,58,141),(71,245,178,250),(72,211,179,251),(73,212,180,252),(74,213,181,253),(75,214,182,254),(76,215,183,255),(77,216,184,256),(78,217,185,257),(79,218,186,258),(80,219,187,259),(81,220,188,260),(82,221,189,261),(83,222,190,262),(84,223,191,263),(85,224,192,264),(86,225,193,265),(87,226,194,266),(88,227,195,267),(89,228,196,268),(90,229,197,269),(91,230,198,270),(92,231,199,271),(93,232,200,272),(94,233,201,273),(95,234,202,274),(96,235,203,275),(97,236,204,276),(98,237,205,277),(99,238,206,278),(100,239,207,279),(101,240,208,280),(102,241,209,246),(103,242,210,247),(104,243,176,248),(105,244,177,249)], [(1,186,59,79),(2,187,60,80),(3,188,61,81),(4,189,62,82),(5,190,63,83),(6,191,64,84),(7,192,65,85),(8,193,66,86),(9,194,67,87),(10,195,68,88),(11,196,69,89),(12,197,70,90),(13,198,36,91),(14,199,37,92),(15,200,38,93),(16,201,39,94),(17,202,40,95),(18,203,41,96),(19,204,42,97),(20,205,43,98),(21,206,44,99),(22,207,45,100),(23,208,46,101),(24,209,47,102),(25,210,48,103),(26,176,49,104),(27,177,50,105),(28,178,51,71),(29,179,52,72),(30,180,53,73),(31,181,54,74),(32,182,55,75),(33,183,56,76),(34,184,57,77),(35,185,58,78),(106,224,148,264),(107,225,149,265),(108,226,150,266),(109,227,151,267),(110,228,152,268),(111,229,153,269),(112,230,154,270),(113,231,155,271),(114,232,156,272),(115,233,157,273),(116,234,158,274),(117,235,159,275),(118,236,160,276),(119,237,161,277),(120,238,162,278),(121,239,163,279),(122,240,164,280),(123,241,165,246),(124,242,166,247),(125,243,167,248),(126,244,168,249),(127,245,169,250),(128,211,170,251),(129,212,171,252),(130,213,172,253),(131,214,173,254),(132,215,174,255),(133,216,175,256),(134,217,141,257),(135,218,142,258),(136,219,143,259),(137,220,144,260),(138,221,145,261),(139,222,146,262),(140,223,147,263)])

175 conjugacy classes

class 1  2 4A4B4C5A5B5C5D7A···7F10A10B10C10D14A···14F20A···20L28A···28R35A···35X70A···70X140A···140BT
order1244455557···71010101014···1420···2028···2835···3570···70140···140
size1122211111···111111···12···22···21···11···12···2

175 irreducible representations

dim111111112222
type++-
imageC1C2C5C7C10C14C35C70Q8C5×Q8C7×Q8Q8×C35
kernelQ8×C35C140C7×Q8C5×Q8C28C20Q8C4C35C7C5C1
# reps13461218247214624

Matrix representation of Q8×C35 in GL2(𝔽281) generated by

2000
0200
,
253279
25228
,
142248
15139
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

Subgroup lattice of Q8×C35 in TeX

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