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G = C453C8order 360 = 23·32·5

1st semidirect product of C45 and C8 acting via C8/C4=C2

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

Aliases: C453C8, C90.3C4, C60.4S3, C4.2D45, C20.2D9, C36.2D5, C18.Dic5, C2.Dic45, C180.2C2, C12.4D15, C30.5Dic3, C6.1Dic15, C10.2Dic9, C52(C9⋊C8), C9⋊(C52C8), C15.2(C3⋊C8), C3.(C153C8), SmallGroup(360,3)

Series: Derived Chief Lower central Upper central

C1C45 — C453C8
C1C3C15C45C90C180 — C453C8
C45 — C453C8
C1C4

Generators and relations for C453C8
 G = < a,b | a45=b8=1, bab-1=a-1 >

45C8
15C3⋊C8
9C52C8
5C9⋊C8
3C153C8

Smallest permutation representation of C453C8
Regular action on 360 points
Generators in S360
(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)(271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315)(316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360)
(1 342 158 239 90 273 119 188)(2 341 159 238 46 272 120 187)(3 340 160 237 47 271 121 186)(4 339 161 236 48 315 122 185)(5 338 162 235 49 314 123 184)(6 337 163 234 50 313 124 183)(7 336 164 233 51 312 125 182)(8 335 165 232 52 311 126 181)(9 334 166 231 53 310 127 225)(10 333 167 230 54 309 128 224)(11 332 168 229 55 308 129 223)(12 331 169 228 56 307 130 222)(13 330 170 227 57 306 131 221)(14 329 171 226 58 305 132 220)(15 328 172 270 59 304 133 219)(16 327 173 269 60 303 134 218)(17 326 174 268 61 302 135 217)(18 325 175 267 62 301 91 216)(19 324 176 266 63 300 92 215)(20 323 177 265 64 299 93 214)(21 322 178 264 65 298 94 213)(22 321 179 263 66 297 95 212)(23 320 180 262 67 296 96 211)(24 319 136 261 68 295 97 210)(25 318 137 260 69 294 98 209)(26 317 138 259 70 293 99 208)(27 316 139 258 71 292 100 207)(28 360 140 257 72 291 101 206)(29 359 141 256 73 290 102 205)(30 358 142 255 74 289 103 204)(31 357 143 254 75 288 104 203)(32 356 144 253 76 287 105 202)(33 355 145 252 77 286 106 201)(34 354 146 251 78 285 107 200)(35 353 147 250 79 284 108 199)(36 352 148 249 80 283 109 198)(37 351 149 248 81 282 110 197)(38 350 150 247 82 281 111 196)(39 349 151 246 83 280 112 195)(40 348 152 245 84 279 113 194)(41 347 153 244 85 278 114 193)(42 346 154 243 86 277 115 192)(43 345 155 242 87 276 116 191)(44 344 156 241 88 275 117 190)(45 343 157 240 89 274 118 189)

G:=sub<Sym(360)| (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)(271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315)(316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360), (1,342,158,239,90,273,119,188)(2,341,159,238,46,272,120,187)(3,340,160,237,47,271,121,186)(4,339,161,236,48,315,122,185)(5,338,162,235,49,314,123,184)(6,337,163,234,50,313,124,183)(7,336,164,233,51,312,125,182)(8,335,165,232,52,311,126,181)(9,334,166,231,53,310,127,225)(10,333,167,230,54,309,128,224)(11,332,168,229,55,308,129,223)(12,331,169,228,56,307,130,222)(13,330,170,227,57,306,131,221)(14,329,171,226,58,305,132,220)(15,328,172,270,59,304,133,219)(16,327,173,269,60,303,134,218)(17,326,174,268,61,302,135,217)(18,325,175,267,62,301,91,216)(19,324,176,266,63,300,92,215)(20,323,177,265,64,299,93,214)(21,322,178,264,65,298,94,213)(22,321,179,263,66,297,95,212)(23,320,180,262,67,296,96,211)(24,319,136,261,68,295,97,210)(25,318,137,260,69,294,98,209)(26,317,138,259,70,293,99,208)(27,316,139,258,71,292,100,207)(28,360,140,257,72,291,101,206)(29,359,141,256,73,290,102,205)(30,358,142,255,74,289,103,204)(31,357,143,254,75,288,104,203)(32,356,144,253,76,287,105,202)(33,355,145,252,77,286,106,201)(34,354,146,251,78,285,107,200)(35,353,147,250,79,284,108,199)(36,352,148,249,80,283,109,198)(37,351,149,248,81,282,110,197)(38,350,150,247,82,281,111,196)(39,349,151,246,83,280,112,195)(40,348,152,245,84,279,113,194)(41,347,153,244,85,278,114,193)(42,346,154,243,86,277,115,192)(43,345,155,242,87,276,116,191)(44,344,156,241,88,275,117,190)(45,343,157,240,89,274,118,189)>;

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)(271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315)(316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360), (1,342,158,239,90,273,119,188)(2,341,159,238,46,272,120,187)(3,340,160,237,47,271,121,186)(4,339,161,236,48,315,122,185)(5,338,162,235,49,314,123,184)(6,337,163,234,50,313,124,183)(7,336,164,233,51,312,125,182)(8,335,165,232,52,311,126,181)(9,334,166,231,53,310,127,225)(10,333,167,230,54,309,128,224)(11,332,168,229,55,308,129,223)(12,331,169,228,56,307,130,222)(13,330,170,227,57,306,131,221)(14,329,171,226,58,305,132,220)(15,328,172,270,59,304,133,219)(16,327,173,269,60,303,134,218)(17,326,174,268,61,302,135,217)(18,325,175,267,62,301,91,216)(19,324,176,266,63,300,92,215)(20,323,177,265,64,299,93,214)(21,322,178,264,65,298,94,213)(22,321,179,263,66,297,95,212)(23,320,180,262,67,296,96,211)(24,319,136,261,68,295,97,210)(25,318,137,260,69,294,98,209)(26,317,138,259,70,293,99,208)(27,316,139,258,71,292,100,207)(28,360,140,257,72,291,101,206)(29,359,141,256,73,290,102,205)(30,358,142,255,74,289,103,204)(31,357,143,254,75,288,104,203)(32,356,144,253,76,287,105,202)(33,355,145,252,77,286,106,201)(34,354,146,251,78,285,107,200)(35,353,147,250,79,284,108,199)(36,352,148,249,80,283,109,198)(37,351,149,248,81,282,110,197)(38,350,150,247,82,281,111,196)(39,349,151,246,83,280,112,195)(40,348,152,245,84,279,113,194)(41,347,153,244,85,278,114,193)(42,346,154,243,86,277,115,192)(43,345,155,242,87,276,116,191)(44,344,156,241,88,275,117,190)(45,343,157,240,89,274,118,189) );

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),(271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315),(316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360)], [(1,342,158,239,90,273,119,188),(2,341,159,238,46,272,120,187),(3,340,160,237,47,271,121,186),(4,339,161,236,48,315,122,185),(5,338,162,235,49,314,123,184),(6,337,163,234,50,313,124,183),(7,336,164,233,51,312,125,182),(8,335,165,232,52,311,126,181),(9,334,166,231,53,310,127,225),(10,333,167,230,54,309,128,224),(11,332,168,229,55,308,129,223),(12,331,169,228,56,307,130,222),(13,330,170,227,57,306,131,221),(14,329,171,226,58,305,132,220),(15,328,172,270,59,304,133,219),(16,327,173,269,60,303,134,218),(17,326,174,268,61,302,135,217),(18,325,175,267,62,301,91,216),(19,324,176,266,63,300,92,215),(20,323,177,265,64,299,93,214),(21,322,178,264,65,298,94,213),(22,321,179,263,66,297,95,212),(23,320,180,262,67,296,96,211),(24,319,136,261,68,295,97,210),(25,318,137,260,69,294,98,209),(26,317,138,259,70,293,99,208),(27,316,139,258,71,292,100,207),(28,360,140,257,72,291,101,206),(29,359,141,256,73,290,102,205),(30,358,142,255,74,289,103,204),(31,357,143,254,75,288,104,203),(32,356,144,253,76,287,105,202),(33,355,145,252,77,286,106,201),(34,354,146,251,78,285,107,200),(35,353,147,250,79,284,108,199),(36,352,148,249,80,283,109,198),(37,351,149,248,81,282,110,197),(38,350,150,247,82,281,111,196),(39,349,151,246,83,280,112,195),(40,348,152,245,84,279,113,194),(41,347,153,244,85,278,114,193),(42,346,154,243,86,277,115,192),(43,345,155,242,87,276,116,191),(44,344,156,241,88,275,117,190),(45,343,157,240,89,274,118,189)])

96 conjugacy classes

class 1  2  3 4A4B5A5B 6 8A8B8C8D9A9B9C10A10B12A12B15A15B15C15D18A18B18C20A20B20C20D30A30B30C30D36A···36F45A···45L60A···60H90A···90L180A···180X
order1234455688889991010121215151515181818202020203030303036···3645···4560···6090···90180···180
size112112224545454522222222222222222222222···22···22···22···22···2

96 irreducible representations

dim1111222222222222222
type++++-+-+--+-
imageC1C2C4C8S3D5Dic3D9Dic5C3⋊C8D15Dic9C52C8Dic15C9⋊C8D45C153C8Dic45C453C8
kernelC453C8C180C90C45C60C36C30C20C18C15C12C10C9C6C5C4C3C2C1
# reps1124121322434461281224

Matrix representation of C453C8 in GL2(𝔽1801) generated by

330920
8811250
,
137940
10771664
G:=sub<GL(2,GF(1801))| [330,881,920,1250],[137,1077,940,1664] >;

C453C8 in GAP, Magma, Sage, TeX

C_{45}\rtimes_3C_8
% in TeX

G:=Group("C45:3C8");
// GroupNames label

G:=SmallGroup(360,3);
// by ID

G=gap.SmallGroup(360,3);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-5,-3,12,31,3267,741,2884,8645]);
// Polycyclic

G:=Group<a,b|a^45=b^8=1,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C453C8 in TeX

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