metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C36⋊1C8, C36.7Q8, C4.16D36, C36.32D4, C42.2D9, C12.51D12, C4.7Dic18, C12.26Dic6, C18.5M4(2), C4⋊(C9⋊C8), C9⋊1(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
Generators and relations for C36⋊C8
G = < a,b | a36=b8=1, bab-1=a-1 >
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 71 244 115 259 173 192 86)(2 70 245 114 260 172 193 85)(3 69 246 113 261 171 194 84)(4 68 247 112 262 170 195 83)(5 67 248 111 263 169 196 82)(6 66 249 110 264 168 197 81)(7 65 250 109 265 167 198 80)(8 64 251 144 266 166 199 79)(9 63 252 143 267 165 200 78)(10 62 217 142 268 164 201 77)(11 61 218 141 269 163 202 76)(12 60 219 140 270 162 203 75)(13 59 220 139 271 161 204 74)(14 58 221 138 272 160 205 73)(15 57 222 137 273 159 206 108)(16 56 223 136 274 158 207 107)(17 55 224 135 275 157 208 106)(18 54 225 134 276 156 209 105)(19 53 226 133 277 155 210 104)(20 52 227 132 278 154 211 103)(21 51 228 131 279 153 212 102)(22 50 229 130 280 152 213 101)(23 49 230 129 281 151 214 100)(24 48 231 128 282 150 215 99)(25 47 232 127 283 149 216 98)(26 46 233 126 284 148 181 97)(27 45 234 125 285 147 182 96)(28 44 235 124 286 146 183 95)(29 43 236 123 287 145 184 94)(30 42 237 122 288 180 185 93)(31 41 238 121 253 179 186 92)(32 40 239 120 254 178 187 91)(33 39 240 119 255 177 188 90)(34 38 241 118 256 176 189 89)(35 37 242 117 257 175 190 88)(36 72 243 116 258 174 191 87)
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), (1,71,244,115,259,173,192,86)(2,70,245,114,260,172,193,85)(3,69,246,113,261,171,194,84)(4,68,247,112,262,170,195,83)(5,67,248,111,263,169,196,82)(6,66,249,110,264,168,197,81)(7,65,250,109,265,167,198,80)(8,64,251,144,266,166,199,79)(9,63,252,143,267,165,200,78)(10,62,217,142,268,164,201,77)(11,61,218,141,269,163,202,76)(12,60,219,140,270,162,203,75)(13,59,220,139,271,161,204,74)(14,58,221,138,272,160,205,73)(15,57,222,137,273,159,206,108)(16,56,223,136,274,158,207,107)(17,55,224,135,275,157,208,106)(18,54,225,134,276,156,209,105)(19,53,226,133,277,155,210,104)(20,52,227,132,278,154,211,103)(21,51,228,131,279,153,212,102)(22,50,229,130,280,152,213,101)(23,49,230,129,281,151,214,100)(24,48,231,128,282,150,215,99)(25,47,232,127,283,149,216,98)(26,46,233,126,284,148,181,97)(27,45,234,125,285,147,182,96)(28,44,235,124,286,146,183,95)(29,43,236,123,287,145,184,94)(30,42,237,122,288,180,185,93)(31,41,238,121,253,179,186,92)(32,40,239,120,254,178,187,91)(33,39,240,119,255,177,188,90)(34,38,241,118,256,176,189,89)(35,37,242,117,257,175,190,88)(36,72,243,116,258,174,191,87)>;
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), (1,71,244,115,259,173,192,86)(2,70,245,114,260,172,193,85)(3,69,246,113,261,171,194,84)(4,68,247,112,262,170,195,83)(5,67,248,111,263,169,196,82)(6,66,249,110,264,168,197,81)(7,65,250,109,265,167,198,80)(8,64,251,144,266,166,199,79)(9,63,252,143,267,165,200,78)(10,62,217,142,268,164,201,77)(11,61,218,141,269,163,202,76)(12,60,219,140,270,162,203,75)(13,59,220,139,271,161,204,74)(14,58,221,138,272,160,205,73)(15,57,222,137,273,159,206,108)(16,56,223,136,274,158,207,107)(17,55,224,135,275,157,208,106)(18,54,225,134,276,156,209,105)(19,53,226,133,277,155,210,104)(20,52,227,132,278,154,211,103)(21,51,228,131,279,153,212,102)(22,50,229,130,280,152,213,101)(23,49,230,129,281,151,214,100)(24,48,231,128,282,150,215,99)(25,47,232,127,283,149,216,98)(26,46,233,126,284,148,181,97)(27,45,234,125,285,147,182,96)(28,44,235,124,286,146,183,95)(29,43,236,123,287,145,184,94)(30,42,237,122,288,180,185,93)(31,41,238,121,253,179,186,92)(32,40,239,120,254,178,187,91)(33,39,240,119,255,177,188,90)(34,38,241,118,256,176,189,89)(35,37,242,117,257,175,190,88)(36,72,243,116,258,174,191,87) );
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)], [(1,71,244,115,259,173,192,86),(2,70,245,114,260,172,193,85),(3,69,246,113,261,171,194,84),(4,68,247,112,262,170,195,83),(5,67,248,111,263,169,196,82),(6,66,249,110,264,168,197,81),(7,65,250,109,265,167,198,80),(8,64,251,144,266,166,199,79),(9,63,252,143,267,165,200,78),(10,62,217,142,268,164,201,77),(11,61,218,141,269,163,202,76),(12,60,219,140,270,162,203,75),(13,59,220,139,271,161,204,74),(14,58,221,138,272,160,205,73),(15,57,222,137,273,159,206,108),(16,56,223,136,274,158,207,107),(17,55,224,135,275,157,208,106),(18,54,225,134,276,156,209,105),(19,53,226,133,277,155,210,104),(20,52,227,132,278,154,211,103),(21,51,228,131,279,153,212,102),(22,50,229,130,280,152,213,101),(23,49,230,129,281,151,214,100),(24,48,231,128,282,150,215,99),(25,47,232,127,283,149,216,98),(26,46,233,126,284,148,181,97),(27,45,234,125,285,147,182,96),(28,44,235,124,286,146,183,95),(29,43,236,123,287,145,184,94),(30,42,237,122,288,180,185,93),(31,41,238,121,253,179,186,92),(32,40,239,120,254,178,187,91),(33,39,240,119,255,177,188,90),(34,38,241,118,256,176,189,89),(35,37,242,117,257,175,190,88),(36,72,243,116,258,174,191,87)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 8A | ··· | 8H | 9A | 9B | 9C | 12A | ··· | 12L | 18A | ··· | 18I | 36A | ··· | 36AJ |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 9 | 9 | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 18 | ··· | 18 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | - | + | + | - | + | - | + | - | + | |||||||
| image | C1 | C2 | C2 | C4 | C8 | S3 | D4 | Q8 | Dic3 | D6 | M4(2) | D9 | C3⋊C8 | Dic6 | D12 | Dic9 | D18 | C4.Dic3 | C9⋊C8 | Dic18 | D36 | C4.Dic9 |
| kernel | C36⋊C8 | C2×C9⋊C8 | C4×C36 | C2×C36 | C36 | C4×C12 | C36 | C36 | C2×C12 | C2×C12 | C18 | C42 | C12 | C12 | C12 | C2×C4 | C2×C4 | C6 | C4 | C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 4 | 8 | 1 | 1 | 1 | 2 | 1 | 2 | 3 | 4 | 2 | 2 | 6 | 3 | 4 | 12 | 6 | 6 | 12 |
Matrix representation of C36⋊C8 ►in GL3(𝔽73) generated by
| 72 | 0 | 0 |
| 0 | 48 | 29 |
| 0 | 44 | 19 |
| 63 | 0 | 0 |
| 0 | 7 | 32 |
| 0 | 25 | 66 |
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
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