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G = C31⋊C9order 279 = 32·31

The semidirect product of C31 and C9 acting via C9/C3=C3

metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary

Aliases: C31⋊C9, C93.C3, C3.(C31⋊C3), SmallGroup(279,1)

Series: Derived Chief Lower central Upper central

C1C31 — C31⋊C9
C1C31C93 — C31⋊C9
C31 — C31⋊C9
C1C3

Generators and relations for C31⋊C9
 G = < a,b | a31=b9=1, bab-1=a5 >

31C9

Smallest permutation representation of C31⋊C9
Regular action on 279 points
Generators in S279
(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)
(1 255 160 63 242 155 59 200 95)(2 249 165 64 236 129 60 194 100)(3 274 170 65 230 134 61 188 105)(4 268 175 66 224 139 62 213 110)(5 262 180 67 218 144 32 207 115)(6 256 185 68 243 149 33 201 120)(7 250 159 69 237 154 34 195 94)(8 275 164 70 231 128 35 189 99)(9 269 169 71 225 133 36 214 104)(10 263 174 72 219 138 37 208 109)(11 257 179 73 244 143 38 202 114)(12 251 184 74 238 148 39 196 119)(13 276 158 75 232 153 40 190 124)(14 270 163 76 226 127 41 215 98)(15 264 168 77 220 132 42 209 103)(16 258 173 78 245 137 43 203 108)(17 252 178 79 239 142 44 197 113)(18 277 183 80 233 147 45 191 118)(19 271 157 81 227 152 46 216 123)(20 265 162 82 221 126 47 210 97)(21 259 167 83 246 131 48 204 102)(22 253 172 84 240 136 49 198 107)(23 278 177 85 234 141 50 192 112)(24 272 182 86 228 146 51 217 117)(25 266 156 87 222 151 52 211 122)(26 260 161 88 247 125 53 205 96)(27 254 166 89 241 130 54 199 101)(28 279 171 90 235 135 55 193 106)(29 273 176 91 229 140 56 187 111)(30 267 181 92 223 145 57 212 116)(31 261 186 93 248 150 58 206 121)

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

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

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

39 conjugacy classes

class 1 3A3B9A···9F31A···31J93A···93T
order1339···931···3193···93
size11131···313···33···3

39 irreducible representations

dim11133
type+
imageC1C3C9C31⋊C3C31⋊C9
kernelC31⋊C9C93C31C3C1
# reps1261020

Matrix representation of C31⋊C9 in GL4(𝔽1117) generated by

1000
0010
0001
01160588
,
777000
07706041108
0561828238
0958690636
G:=sub<GL(4,GF(1117))| [1,0,0,0,0,0,0,1,0,1,0,160,0,0,1,588],[777,0,0,0,0,770,561,958,0,604,828,690,0,1108,238,636] >;

C31⋊C9 in GAP, Magma, Sage, TeX

C_{31}\rtimes C_9
% in TeX

G:=Group("C31:C9");
// GroupNames label

G:=SmallGroup(279,1);
// by ID

G=gap.SmallGroup(279,1);
# by ID

G:=PCGroup([3,-3,-3,-31,9,2027]);
// Polycyclic

G:=Group<a,b|a^31=b^9=1,b*a*b^-1=a^5>;
// generators/relations

Export

Subgroup lattice of C31⋊C9 in TeX

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