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G = C353C8order 280 = 23·5·7

1st semidirect product of C35 and C8 acting via C8/C4=C2

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C353C8, C70.3C4, C28.2D5, C4.2D35, C20.2D7, C14.Dic5, C2.Dic35, C140.2C2, C10.2Dic7, C7⋊(C52C8), C52(C7⋊C8), SmallGroup(280,3)

Series: Derived Chief Lower central Upper central

C1C35 — C353C8
C1C7C35C70C140 — C353C8
C35 — C353C8
C1C4

Generators and relations for C353C8
 G = < a,b | a35=b8=1, bab-1=a-1 >

35C8
7C52C8
5C7⋊C8

Smallest permutation representation of C353C8
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)
(1 279 120 193 52 217 85 174)(2 278 121 192 53 216 86 173)(3 277 122 191 54 215 87 172)(4 276 123 190 55 214 88 171)(5 275 124 189 56 213 89 170)(6 274 125 188 57 212 90 169)(7 273 126 187 58 211 91 168)(8 272 127 186 59 245 92 167)(9 271 128 185 60 244 93 166)(10 270 129 184 61 243 94 165)(11 269 130 183 62 242 95 164)(12 268 131 182 63 241 96 163)(13 267 132 181 64 240 97 162)(14 266 133 180 65 239 98 161)(15 265 134 179 66 238 99 160)(16 264 135 178 67 237 100 159)(17 263 136 177 68 236 101 158)(18 262 137 176 69 235 102 157)(19 261 138 210 70 234 103 156)(20 260 139 209 36 233 104 155)(21 259 140 208 37 232 105 154)(22 258 106 207 38 231 71 153)(23 257 107 206 39 230 72 152)(24 256 108 205 40 229 73 151)(25 255 109 204 41 228 74 150)(26 254 110 203 42 227 75 149)(27 253 111 202 43 226 76 148)(28 252 112 201 44 225 77 147)(29 251 113 200 45 224 78 146)(30 250 114 199 46 223 79 145)(31 249 115 198 47 222 80 144)(32 248 116 197 48 221 81 143)(33 247 117 196 49 220 82 142)(34 246 118 195 50 219 83 141)(35 280 119 194 51 218 84 175)

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

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

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

76 conjugacy classes

class 1  2 4A4B5A5B7A7B7C8A8B8C8D10A10B14A14B14C20A20B20C20D28A···28F35A···35L70A···70L140A···140X
order124455777888810101414142020202028···2835···3570···70140···140
size111122222353535352222222222···22···22···22···2

76 irreducible representations

dim1111222222222
type++++--+-
imageC1C2C4C8D5D7Dic5Dic7C52C8C7⋊C8D35Dic35C353C8
kernelC353C8C140C70C35C28C20C14C10C7C5C4C2C1
# reps1124232346121224

Matrix representation of C353C8 in GL3(𝔽281) generated by

100
04419
026294
,
8900
020612
014075
G:=sub<GL(3,GF(281))| [1,0,0,0,44,262,0,19,94],[89,0,0,0,206,140,0,12,75] >;

C353C8 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

Subgroup lattice of C353C8 in TeX

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