metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C72⋊1C4, C8⋊1Dic9, C18.4D8, C6.4D24, C2.1D72, C36.5Q8, C18.2Q16, C24.2Dic3, C2.2Dic36, C4.5Dic18, C6.2Dic12, C22.9D36, C12.20Dic6, (C2×C8).3D9, C9⋊2(C2.D8), (C2×C24).9S3, (C2×C72).5C2, C18.7(C4⋊C4), C3.(C24⋊1C4), C36.37(C2×C4), (C2×C4).75D18, (C2×C18).14D4, (C2×C6).22D12, C4⋊Dic9.3C2, C4.8(C2×Dic9), (C2×C12).364D6, C2.5(C4⋊Dic9), C6.9(C4⋊Dic3), (C2×C36).83C22, C12.43(C2×Dic3), SmallGroup(288,26)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C72⋊1C4
G = < a,b | a72=b4=1, bab-1=a-1 >
Smallest permutation representation of C72⋊1C4
►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)
(1 250 155 120)(2 249 156 119)(3 248 157 118)(4 247 158 117)(5 246 159 116)(6 245 160 115)(7 244 161 114)(8 243 162 113)(9 242 163 112)(10 241 164 111)(11 240 165 110)(12 239 166 109)(13 238 167 108)(14 237 168 107)(15 236 169 106)(16 235 170 105)(17 234 171 104)(18 233 172 103)(19 232 173 102)(20 231 174 101)(21 230 175 100)(22 229 176 99)(23 228 177 98)(24 227 178 97)(25 226 179 96)(26 225 180 95)(27 224 181 94)(28 223 182 93)(29 222 183 92)(30 221 184 91)(31 220 185 90)(32 219 186 89)(33 218 187 88)(34 217 188 87)(35 288 189 86)(36 287 190 85)(37 286 191 84)(38 285 192 83)(39 284 193 82)(40 283 194 81)(41 282 195 80)(42 281 196 79)(43 280 197 78)(44 279 198 77)(45 278 199 76)(46 277 200 75)(47 276 201 74)(48 275 202 73)(49 274 203 144)(50 273 204 143)(51 272 205 142)(52 271 206 141)(53 270 207 140)(54 269 208 139)(55 268 209 138)(56 267 210 137)(57 266 211 136)(58 265 212 135)(59 264 213 134)(60 263 214 133)(61 262 215 132)(62 261 216 131)(63 260 145 130)(64 259 146 129)(65 258 147 128)(66 257 148 127)(67 256 149 126)(68 255 150 125)(69 254 151 124)(70 253 152 123)(71 252 153 122)(72 251 154 121)
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), (1,250,155,120)(2,249,156,119)(3,248,157,118)(4,247,158,117)(5,246,159,116)(6,245,160,115)(7,244,161,114)(8,243,162,113)(9,242,163,112)(10,241,164,111)(11,240,165,110)(12,239,166,109)(13,238,167,108)(14,237,168,107)(15,236,169,106)(16,235,170,105)(17,234,171,104)(18,233,172,103)(19,232,173,102)(20,231,174,101)(21,230,175,100)(22,229,176,99)(23,228,177,98)(24,227,178,97)(25,226,179,96)(26,225,180,95)(27,224,181,94)(28,223,182,93)(29,222,183,92)(30,221,184,91)(31,220,185,90)(32,219,186,89)(33,218,187,88)(34,217,188,87)(35,288,189,86)(36,287,190,85)(37,286,191,84)(38,285,192,83)(39,284,193,82)(40,283,194,81)(41,282,195,80)(42,281,196,79)(43,280,197,78)(44,279,198,77)(45,278,199,76)(46,277,200,75)(47,276,201,74)(48,275,202,73)(49,274,203,144)(50,273,204,143)(51,272,205,142)(52,271,206,141)(53,270,207,140)(54,269,208,139)(55,268,209,138)(56,267,210,137)(57,266,211,136)(58,265,212,135)(59,264,213,134)(60,263,214,133)(61,262,215,132)(62,261,216,131)(63,260,145,130)(64,259,146,129)(65,258,147,128)(66,257,148,127)(67,256,149,126)(68,255,150,125)(69,254,151,124)(70,253,152,123)(71,252,153,122)(72,251,154,121)>;
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), (1,250,155,120)(2,249,156,119)(3,248,157,118)(4,247,158,117)(5,246,159,116)(6,245,160,115)(7,244,161,114)(8,243,162,113)(9,242,163,112)(10,241,164,111)(11,240,165,110)(12,239,166,109)(13,238,167,108)(14,237,168,107)(15,236,169,106)(16,235,170,105)(17,234,171,104)(18,233,172,103)(19,232,173,102)(20,231,174,101)(21,230,175,100)(22,229,176,99)(23,228,177,98)(24,227,178,97)(25,226,179,96)(26,225,180,95)(27,224,181,94)(28,223,182,93)(29,222,183,92)(30,221,184,91)(31,220,185,90)(32,219,186,89)(33,218,187,88)(34,217,188,87)(35,288,189,86)(36,287,190,85)(37,286,191,84)(38,285,192,83)(39,284,193,82)(40,283,194,81)(41,282,195,80)(42,281,196,79)(43,280,197,78)(44,279,198,77)(45,278,199,76)(46,277,200,75)(47,276,201,74)(48,275,202,73)(49,274,203,144)(50,273,204,143)(51,272,205,142)(52,271,206,141)(53,270,207,140)(54,269,208,139)(55,268,209,138)(56,267,210,137)(57,266,211,136)(58,265,212,135)(59,264,213,134)(60,263,214,133)(61,262,215,132)(62,261,216,131)(63,260,145,130)(64,259,146,129)(65,258,147,128)(66,257,148,127)(67,256,149,126)(68,255,150,125)(69,254,151,124)(70,253,152,123)(71,252,153,122)(72,251,154,121) );
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)], [(1,250,155,120),(2,249,156,119),(3,248,157,118),(4,247,158,117),(5,246,159,116),(6,245,160,115),(7,244,161,114),(8,243,162,113),(9,242,163,112),(10,241,164,111),(11,240,165,110),(12,239,166,109),(13,238,167,108),(14,237,168,107),(15,236,169,106),(16,235,170,105),(17,234,171,104),(18,233,172,103),(19,232,173,102),(20,231,174,101),(21,230,175,100),(22,229,176,99),(23,228,177,98),(24,227,178,97),(25,226,179,96),(26,225,180,95),(27,224,181,94),(28,223,182,93),(29,222,183,92),(30,221,184,91),(31,220,185,90),(32,219,186,89),(33,218,187,88),(34,217,188,87),(35,288,189,86),(36,287,190,85),(37,286,191,84),(38,285,192,83),(39,284,193,82),(40,283,194,81),(41,282,195,80),(42,281,196,79),(43,280,197,78),(44,279,198,77),(45,278,199,76),(46,277,200,75),(47,276,201,74),(48,275,202,73),(49,274,203,144),(50,273,204,143),(51,272,205,142),(52,271,206,141),(53,270,207,140),(54,269,208,139),(55,268,209,138),(56,267,210,137),(57,266,211,136),(58,265,212,135),(59,264,213,134),(60,263,214,133),(61,262,215,132),(62,261,216,131),(63,260,145,130),(64,259,146,129),(65,258,147,128),(66,257,148,127),(67,256,149,126),(68,255,150,125),(69,254,151,124),(70,253,152,123),(71,252,153,122),(72,251,154,121)]])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 18A | ··· | 18I | 24A | ··· | 24H | 36A | ··· | 36L | 72A | ··· | 72X |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 36 | 36 | 36 | 36 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | + | - | + | - | + | - | + | + | - | - | + | - | + | |
| image | C1 | C2 | C2 | C4 | S3 | Q8 | D4 | Dic3 | D6 | D8 | Q16 | D9 | Dic6 | D12 | Dic9 | D18 | D24 | Dic12 | Dic18 | D36 | Dic36 | D72 |
| kernel | C72⋊1C4 | C4⋊Dic9 | C2×C72 | C72 | C2×C24 | C36 | C2×C18 | C24 | C2×C12 | C18 | C18 | C2×C8 | C12 | C2×C6 | C8 | C2×C4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 3 | 2 | 2 | 6 | 3 | 4 | 4 | 6 | 6 | 12 | 12 |
Matrix representation of C72⋊1C4 ►in GL3(𝔽73) generated by
| 1 | 0 | 0 |
| 0 | 11 | 13 |
| 0 | 60 | 71 |
| 46 | 0 | 0 |
| 0 | 18 | 68 |
| 0 | 50 | 55 |
G:=sub<GL(3,GF(73))| [1,0,0,0,11,60,0,13,71],[46,0,0,0,18,50,0,68,55] >;
C72⋊1C4 in GAP, Magma, Sage, TeX
C_{72}\rtimes_1C_4 % in TeX
G:=Group("C72:1C4"); // GroupNames label
G:=SmallGroup(288,26);
// by ID
G=gap.SmallGroup(288,26);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,176,675,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b|a^72=b^4=1,b*a*b^-1=a^-1>;
// generators/relations
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