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G = C36⋊C8order 288 = 25·32

1st semidirect product of C36 and C8 acting via C8/C4=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C361C8, C36.7Q8, C4.16D36, C36.32D4, C42.2D9, C12.51D12, C4.7Dic18, C12.26Dic6, C18.5M4(2), C4⋊(C9⋊C8), C91(C4⋊C8), C12.1(C3⋊C8), C18.7(C2×C8), (C4×C36).4C2, (C2×C36).4C4, C3.(C12⋊C8), C18.1(C4⋊C4), (C4×C12).10S3, (C2×C4).89D18, (C2×C4).3Dic9, (C2×C12).403D6, C2.1(C4⋊Dic9), C6.6(C4⋊Dic3), (C2×C12).6Dic3, C6.5(C4.Dic3), C2.2(C4.Dic9), C22.8(C2×Dic9), (C2×C36).101C22, C6.7(C2×C3⋊C8), C2.3(C2×C9⋊C8), (C2×C9⋊C8).8C2, (C2×C18).26(C2×C4), (C2×C6).30(C2×Dic3), SmallGroup(288,11)

Series: Derived Chief Lower central Upper central

C1C18 — C36⋊C8
C1C3C9C18C36C2×C36C2×C9⋊C8 — C36⋊C8
C9C18 — C36⋊C8
C1C2×C4C42

Generators and relations for C36⋊C8
 G = < a,b | a36=b8=1, bab-1=a-1 >

2C4
18C8
18C8
2C12
9C2×C8
9C2×C8
6C3⋊C8
6C3⋊C8
2C36
9C4⋊C8
3C2×C3⋊C8
3C2×C3⋊C8
2C9⋊C8
2C9⋊C8
3C12⋊C8

Smallest permutation representation of C36⋊C8
Regular action on 288 points
Generators in S288
(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)
(1 92 265 252 174 44 203 127)(2 91 266 251 175 43 204 126)(3 90 267 250 176 42 205 125)(4 89 268 249 177 41 206 124)(5 88 269 248 178 40 207 123)(6 87 270 247 179 39 208 122)(7 86 271 246 180 38 209 121)(8 85 272 245 145 37 210 120)(9 84 273 244 146 72 211 119)(10 83 274 243 147 71 212 118)(11 82 275 242 148 70 213 117)(12 81 276 241 149 69 214 116)(13 80 277 240 150 68 215 115)(14 79 278 239 151 67 216 114)(15 78 279 238 152 66 181 113)(16 77 280 237 153 65 182 112)(17 76 281 236 154 64 183 111)(18 75 282 235 155 63 184 110)(19 74 283 234 156 62 185 109)(20 73 284 233 157 61 186 144)(21 108 285 232 158 60 187 143)(22 107 286 231 159 59 188 142)(23 106 287 230 160 58 189 141)(24 105 288 229 161 57 190 140)(25 104 253 228 162 56 191 139)(26 103 254 227 163 55 192 138)(27 102 255 226 164 54 193 137)(28 101 256 225 165 53 194 136)(29 100 257 224 166 52 195 135)(30 99 258 223 167 51 196 134)(31 98 259 222 168 50 197 133)(32 97 260 221 169 49 198 132)(33 96 261 220 170 48 199 131)(34 95 262 219 171 47 200 130)(35 94 263 218 172 46 201 129)(36 93 264 217 173 45 202 128)

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

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

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

84 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H6A6B6C8A···8H9A9B9C12A···12L18A···18I36A···36AJ
order12223444444446668···899912···1218···1836···36
size111121111222222218···182222···22···22···2

84 irreducible representations

dim1111122222222222222222
type+++++--++-+-+-+
imageC1C2C2C4C8S3D4Q8Dic3D6M4(2)D9C3⋊C8Dic6D12Dic9D18C4.Dic3C9⋊C8Dic18D36C4.Dic9
kernelC36⋊C8C2×C9⋊C8C4×C36C2×C36C36C4×C12C36C36C2×C12C2×C12C18C42C12C12C12C2×C4C2×C4C6C4C4C4C2
# reps121481112123422634126612

Matrix representation of C36⋊C8 in GL3(𝔽73) generated by

7200
04829
04419
,
6300
0732
02566
G:=sub<GL(3,GF(73))| [72,0,0,0,48,44,0,29,19],[63,0,0,0,7,25,0,32,66] >;

C36⋊C8 in GAP, Magma, Sage, TeX

C_{36}\rtimes C_8
% in TeX

G:=Group("C36:C8");
// GroupNames label

G:=SmallGroup(288,11);
// by ID

G=gap.SmallGroup(288,11);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,64,100,6725,292,9414]);
// Polycyclic

G:=Group<a,b|a^36=b^8=1,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C36⋊C8 in TeX

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