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G = C45⋊C8order 360 = 23·32·5

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

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

Aliases: C451C8, C18.F5, C90.1C4, C10.Dic9, Dic5.2D9, C30.1Dic3, C5⋊(C9⋊C8), C9⋊(C5⋊C8), C2.(C9⋊F5), C15.(C3⋊C8), C6.1(C3⋊F5), C3.(C15⋊C8), (C9×Dic5).2C2, (C3×Dic5).7S3, SmallGroup(360,6)

Series: Derived Chief Lower central Upper central

C1C45 — C45⋊C8
C1C3C15C45C90C9×Dic5 — C45⋊C8
C45 — C45⋊C8
C1C2

Generators and relations for C45⋊C8
 G = < a,b | a45=b8=1, bab-1=a17 >

5C4
45C8
5C12
15C3⋊C8
5C36
9C5⋊C8
5C9⋊C8
3C15⋊C8

Smallest permutation representation of C45⋊C8
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)
(1 337 150 250 62 291 99 185)(2 345 169 267 63 299 118 202)(3 353 143 239 64 307 92 219)(4 316 162 256 65 315 111 191)(5 324 136 228 66 278 130 208)(6 332 155 245 67 286 104 225)(7 340 174 262 68 294 123 197)(8 348 148 234 69 302 97 214)(9 356 167 251 70 310 116 186)(10 319 141 268 71 273 135 203)(11 327 160 240 72 281 109 220)(12 335 179 257 73 289 128 192)(13 343 153 229 74 297 102 209)(14 351 172 246 75 305 121 181)(15 359 146 263 76 313 95 198)(16 322 165 235 77 276 114 215)(17 330 139 252 78 284 133 187)(18 338 158 269 79 292 107 204)(19 346 177 241 80 300 126 221)(20 354 151 258 81 308 100 193)(21 317 170 230 82 271 119 210)(22 325 144 247 83 279 93 182)(23 333 163 264 84 287 112 199)(24 341 137 236 85 295 131 216)(25 349 156 253 86 303 105 188)(26 357 175 270 87 311 124 205)(27 320 149 242 88 274 98 222)(28 328 168 259 89 282 117 194)(29 336 142 231 90 290 91 211)(30 344 161 248 46 298 110 183)(31 352 180 265 47 306 129 200)(32 360 154 237 48 314 103 217)(33 323 173 254 49 277 122 189)(34 331 147 226 50 285 96 206)(35 339 166 243 51 293 115 223)(36 347 140 260 52 301 134 195)(37 355 159 232 53 309 108 212)(38 318 178 249 54 272 127 184)(39 326 152 266 55 280 101 201)(40 334 171 238 56 288 120 218)(41 342 145 255 57 296 94 190)(42 350 164 227 58 304 113 207)(43 358 138 244 59 312 132 224)(44 321 157 261 60 275 106 196)(45 329 176 233 61 283 125 213)

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

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

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

42 conjugacy classes

class 1  2  3 4A4B 5  6 8A8B8C8D9A9B9C 10 12A12B15A15B18A18B18C30A30B36A···36F45A···45F90A···90F
order123445688889991012121515181818303036···3645···4590···90
size11255424545454522241010442224410···104···44···4

42 irreducible representations

dim1111222222444444
type+++-+-+-
imageC1C2C4C8S3Dic3D9C3⋊C8Dic9C9⋊C8F5C5⋊C8C3⋊F5C15⋊C8C9⋊F5C45⋊C8
kernelC45⋊C8C9×Dic5C90C45C3×Dic5C30Dic5C15C10C5C18C9C6C3C2C1
# reps1124113236112266

Matrix representation of C45⋊C8 in GL8(𝔽1801)

15031067000000
734436000000
00180010000
00180000000
0000216461646158
0000131365211481644
0000168713041561148
000016871561304652
,
5241337000000
8131277000000
0017917880000
00778100000
0000172212961196924
00001363701587926
0000122958411271510
0000871044926383

G:=sub<GL(8,GF(1801))| [1503,734,0,0,0,0,0,0,1067,436,0,0,0,0,0,0,0,0,1800,1800,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,2,1313,1687,1687,0,0,0,0,1646,652,1304,156,0,0,0,0,1646,1148,156,1304,0,0,0,0,158,1644,1148,652],[524,813,0,0,0,0,0,0,1337,1277,0,0,0,0,0,0,0,0,1791,778,0,0,0,0,0,0,788,10,0,0,0,0,0,0,0,0,1722,136,1229,87,0,0,0,0,1296,370,584,1044,0,0,0,0,1196,1587,1127,926,0,0,0,0,924,926,1510,383] >;

C45⋊C8 in GAP, Magma, Sage, TeX

C_{45}\rtimes C_8
% in TeX

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

G:=SmallGroup(360,6);
// by ID

G=gap.SmallGroup(360,6);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-5,-3,12,31,3267,741,2164,1450,8645]);
// Polycyclic

G:=Group<a,b|a^45=b^8=1,b*a*b^-1=a^17>;
// generators/relations

Export

Subgroup lattice of C45⋊C8 in TeX

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