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G = C35⋊Q8order 280 = 23·5·7

The semidirect product of C35 and Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C35⋊Q8, C71Dic10, C51Dic14, 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

C1C70 — C35⋊Q8
C1C7C35C70C5×Dic7 — C35⋊Q8
C35C70 — C35⋊Q8
C1C2

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 >

5C4
7C4
35C4
35Q8
7C20
7Dic5
5C28
5Dic7
7Dic10
5Dic14

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

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

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

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

37 conjugacy classes

class 1  2 4A4B4C5A5B7A7B7C10A10B14A14B14C20A20B20C20D28A···28F35A···35F70A···70F
order124445577710101414142020202028···2835···3570···70
size1110147022222222221414141410···104···44···4

37 irreducible representations

dim1111222222244
type++++-++++--+-
imageC1C2C2C2Q8D5D7D10D14Dic10Dic14D5×D7C35⋊Q8
kernelC35⋊Q8C7×Dic5C5×Dic7Dic35C35Dic7Dic5C14C10C7C5C2C1
# reps1111123234666

Matrix representation of C35⋊Q8 in GL4(𝔽281) generated by

0100
28024300
0039241
0080235
,
10415300
13517700
0045156
00250236
,
843400
24719700
00199236
008782
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

Subgroup lattice of C35⋊Q8 in TeX

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