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G = C393C8order 312 = 23·3·13

1st semidirect product of C39 and C8 acting via C8/C4=C2

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

Aliases: C393C8, C78.3C4, C52.2S3, C4.2D39, C6.Dic13, C2.Dic39, C156.2C2, C12.2D13, C26.2Dic3, C3⋊(C132C8), C132(C3⋊C8), SmallGroup(312,5)

Series: Derived Chief Lower central Upper central

C1C39 — C393C8
C1C13C39C78C156 — C393C8
C39 — C393C8
C1C4

Generators and relations for C393C8
 G = < a,b | a39=b8=1, bab-1=a-1 >

39C8
13C3⋊C8
3C132C8

Smallest permutation representation of C393C8
Regular action on 312 points
Generators in S312
(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)(274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312)
(1 293 124 217 68 272 102 194)(2 292 125 216 69 271 103 193)(3 291 126 215 70 270 104 192)(4 290 127 214 71 269 105 191)(5 289 128 213 72 268 106 190)(6 288 129 212 73 267 107 189)(7 287 130 211 74 266 108 188)(8 286 131 210 75 265 109 187)(9 285 132 209 76 264 110 186)(10 284 133 208 77 263 111 185)(11 283 134 207 78 262 112 184)(12 282 135 206 40 261 113 183)(13 281 136 205 41 260 114 182)(14 280 137 204 42 259 115 181)(15 279 138 203 43 258 116 180)(16 278 139 202 44 257 117 179)(17 277 140 201 45 256 79 178)(18 276 141 200 46 255 80 177)(19 275 142 199 47 254 81 176)(20 274 143 198 48 253 82 175)(21 312 144 197 49 252 83 174)(22 311 145 196 50 251 84 173)(23 310 146 234 51 250 85 172)(24 309 147 233 52 249 86 171)(25 308 148 232 53 248 87 170)(26 307 149 231 54 247 88 169)(27 306 150 230 55 246 89 168)(28 305 151 229 56 245 90 167)(29 304 152 228 57 244 91 166)(30 303 153 227 58 243 92 165)(31 302 154 226 59 242 93 164)(32 301 155 225 60 241 94 163)(33 300 156 224 61 240 95 162)(34 299 118 223 62 239 96 161)(35 298 119 222 63 238 97 160)(36 297 120 221 64 237 98 159)(37 296 121 220 65 236 99 158)(38 295 122 219 66 235 100 157)(39 294 123 218 67 273 101 195)

G:=sub<Sym(312)| (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)(274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312), (1,293,124,217,68,272,102,194)(2,292,125,216,69,271,103,193)(3,291,126,215,70,270,104,192)(4,290,127,214,71,269,105,191)(5,289,128,213,72,268,106,190)(6,288,129,212,73,267,107,189)(7,287,130,211,74,266,108,188)(8,286,131,210,75,265,109,187)(9,285,132,209,76,264,110,186)(10,284,133,208,77,263,111,185)(11,283,134,207,78,262,112,184)(12,282,135,206,40,261,113,183)(13,281,136,205,41,260,114,182)(14,280,137,204,42,259,115,181)(15,279,138,203,43,258,116,180)(16,278,139,202,44,257,117,179)(17,277,140,201,45,256,79,178)(18,276,141,200,46,255,80,177)(19,275,142,199,47,254,81,176)(20,274,143,198,48,253,82,175)(21,312,144,197,49,252,83,174)(22,311,145,196,50,251,84,173)(23,310,146,234,51,250,85,172)(24,309,147,233,52,249,86,171)(25,308,148,232,53,248,87,170)(26,307,149,231,54,247,88,169)(27,306,150,230,55,246,89,168)(28,305,151,229,56,245,90,167)(29,304,152,228,57,244,91,166)(30,303,153,227,58,243,92,165)(31,302,154,226,59,242,93,164)(32,301,155,225,60,241,94,163)(33,300,156,224,61,240,95,162)(34,299,118,223,62,239,96,161)(35,298,119,222,63,238,97,160)(36,297,120,221,64,237,98,159)(37,296,121,220,65,236,99,158)(38,295,122,219,66,235,100,157)(39,294,123,218,67,273,101,195)>;

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)(274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312), (1,293,124,217,68,272,102,194)(2,292,125,216,69,271,103,193)(3,291,126,215,70,270,104,192)(4,290,127,214,71,269,105,191)(5,289,128,213,72,268,106,190)(6,288,129,212,73,267,107,189)(7,287,130,211,74,266,108,188)(8,286,131,210,75,265,109,187)(9,285,132,209,76,264,110,186)(10,284,133,208,77,263,111,185)(11,283,134,207,78,262,112,184)(12,282,135,206,40,261,113,183)(13,281,136,205,41,260,114,182)(14,280,137,204,42,259,115,181)(15,279,138,203,43,258,116,180)(16,278,139,202,44,257,117,179)(17,277,140,201,45,256,79,178)(18,276,141,200,46,255,80,177)(19,275,142,199,47,254,81,176)(20,274,143,198,48,253,82,175)(21,312,144,197,49,252,83,174)(22,311,145,196,50,251,84,173)(23,310,146,234,51,250,85,172)(24,309,147,233,52,249,86,171)(25,308,148,232,53,248,87,170)(26,307,149,231,54,247,88,169)(27,306,150,230,55,246,89,168)(28,305,151,229,56,245,90,167)(29,304,152,228,57,244,91,166)(30,303,153,227,58,243,92,165)(31,302,154,226,59,242,93,164)(32,301,155,225,60,241,94,163)(33,300,156,224,61,240,95,162)(34,299,118,223,62,239,96,161)(35,298,119,222,63,238,97,160)(36,297,120,221,64,237,98,159)(37,296,121,220,65,236,99,158)(38,295,122,219,66,235,100,157)(39,294,123,218,67,273,101,195) );

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),(274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312)], [(1,293,124,217,68,272,102,194),(2,292,125,216,69,271,103,193),(3,291,126,215,70,270,104,192),(4,290,127,214,71,269,105,191),(5,289,128,213,72,268,106,190),(6,288,129,212,73,267,107,189),(7,287,130,211,74,266,108,188),(8,286,131,210,75,265,109,187),(9,285,132,209,76,264,110,186),(10,284,133,208,77,263,111,185),(11,283,134,207,78,262,112,184),(12,282,135,206,40,261,113,183),(13,281,136,205,41,260,114,182),(14,280,137,204,42,259,115,181),(15,279,138,203,43,258,116,180),(16,278,139,202,44,257,117,179),(17,277,140,201,45,256,79,178),(18,276,141,200,46,255,80,177),(19,275,142,199,47,254,81,176),(20,274,143,198,48,253,82,175),(21,312,144,197,49,252,83,174),(22,311,145,196,50,251,84,173),(23,310,146,234,51,250,85,172),(24,309,147,233,52,249,86,171),(25,308,148,232,53,248,87,170),(26,307,149,231,54,247,88,169),(27,306,150,230,55,246,89,168),(28,305,151,229,56,245,90,167),(29,304,152,228,57,244,91,166),(30,303,153,227,58,243,92,165),(31,302,154,226,59,242,93,164),(32,301,155,225,60,241,94,163),(33,300,156,224,61,240,95,162),(34,299,118,223,62,239,96,161),(35,298,119,222,63,238,97,160),(36,297,120,221,64,237,98,159),(37,296,121,220,65,236,99,158),(38,295,122,219,66,235,100,157),(39,294,123,218,67,273,101,195)])

84 conjugacy classes

class 1  2  3 4A4B 6 8A8B8C8D12A12B13A···13F26A···26F39A···39L52A···52L78A···78L156A···156X
order1234468888121213···1326···2639···3952···5278···78156···156
size11211239393939222···22···22···22···22···22···2

84 irreducible representations

dim1111222222222
type+++-+-+-
imageC1C2C4C8S3Dic3C3⋊C8D13Dic13D39C132C8Dic39C393C8
kernelC393C8C156C78C39C52C26C13C12C6C4C3C2C1
# reps11241126612121224

Matrix representation of C393C8 in GL2(𝔽313) generated by

22861
25299
,
210215
166103
G:=sub<GL(2,GF(313))| [228,252,61,99],[210,166,215,103] >;

C393C8 in GAP, Magma, Sage, TeX

C_{39}\rtimes_3C_8
% in TeX

G:=Group("C39:3C8");
// GroupNames label

G:=SmallGroup(312,5);
// by ID

G=gap.SmallGroup(312,5);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-13,10,26,323,7204]);
// Polycyclic

G:=Group<a,b|a^39=b^8=1,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C393C8 in TeX

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