metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C72⋊4C4, C8⋊3Dic9, C18.4C42, C24.9Dic3, C18.2M4(2), C9⋊C8⋊4C4, (C2×C8).8D9, C9⋊2(C8⋊C4), C4.21(C4×D9), C3.(C24⋊C4), C12.70(C4×S3), (C2×C24).26S3, C36.26(C2×C4), (C2×C72).12C2, (C2×C4).92D18, C2.4(C4×Dic9), C6.9(C4×Dic3), C6.5(C8⋊S3), (C2×C12).406D6, C2.2(C8⋊D9), (C4×Dic9).6C2, (C2×Dic9).3C4, C4.12(C2×Dic9), C22.10(C4×D9), C12.52(C2×Dic3), (C2×C36).104C22, (C2×C9⋊C8).10C2, (C2×C6).36(C4×S3), (C2×C18).11(C2×C4), SmallGroup(288,23)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C72⋊C4
G = < a,b | a72=b4=1, bab-1=a53 >
Smallest permutation representation of C72⋊C4
►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)
(1 267 90 149)(2 248 91 202)(3 229 92 183)(4 282 93 164)(5 263 94 145)(6 244 95 198)(7 225 96 179)(8 278 97 160)(9 259 98 213)(10 240 99 194)(11 221 100 175)(12 274 101 156)(13 255 102 209)(14 236 103 190)(15 217 104 171)(16 270 105 152)(17 251 106 205)(18 232 107 186)(19 285 108 167)(20 266 109 148)(21 247 110 201)(22 228 111 182)(23 281 112 163)(24 262 113 216)(25 243 114 197)(26 224 115 178)(27 277 116 159)(28 258 117 212)(29 239 118 193)(30 220 119 174)(31 273 120 155)(32 254 121 208)(33 235 122 189)(34 288 123 170)(35 269 124 151)(36 250 125 204)(37 231 126 185)(38 284 127 166)(39 265 128 147)(40 246 129 200)(41 227 130 181)(42 280 131 162)(43 261 132 215)(44 242 133 196)(45 223 134 177)(46 276 135 158)(47 257 136 211)(48 238 137 192)(49 219 138 173)(50 272 139 154)(51 253 140 207)(52 234 141 188)(53 287 142 169)(54 268 143 150)(55 249 144 203)(56 230 73 184)(57 283 74 165)(58 264 75 146)(59 245 76 199)(60 226 77 180)(61 279 78 161)(62 260 79 214)(63 241 80 195)(64 222 81 176)(65 275 82 157)(66 256 83 210)(67 237 84 191)(68 218 85 172)(69 271 86 153)(70 252 87 206)(71 233 88 187)(72 286 89 168)
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), (1,267,90,149)(2,248,91,202)(3,229,92,183)(4,282,93,164)(5,263,94,145)(6,244,95,198)(7,225,96,179)(8,278,97,160)(9,259,98,213)(10,240,99,194)(11,221,100,175)(12,274,101,156)(13,255,102,209)(14,236,103,190)(15,217,104,171)(16,270,105,152)(17,251,106,205)(18,232,107,186)(19,285,108,167)(20,266,109,148)(21,247,110,201)(22,228,111,182)(23,281,112,163)(24,262,113,216)(25,243,114,197)(26,224,115,178)(27,277,116,159)(28,258,117,212)(29,239,118,193)(30,220,119,174)(31,273,120,155)(32,254,121,208)(33,235,122,189)(34,288,123,170)(35,269,124,151)(36,250,125,204)(37,231,126,185)(38,284,127,166)(39,265,128,147)(40,246,129,200)(41,227,130,181)(42,280,131,162)(43,261,132,215)(44,242,133,196)(45,223,134,177)(46,276,135,158)(47,257,136,211)(48,238,137,192)(49,219,138,173)(50,272,139,154)(51,253,140,207)(52,234,141,188)(53,287,142,169)(54,268,143,150)(55,249,144,203)(56,230,73,184)(57,283,74,165)(58,264,75,146)(59,245,76,199)(60,226,77,180)(61,279,78,161)(62,260,79,214)(63,241,80,195)(64,222,81,176)(65,275,82,157)(66,256,83,210)(67,237,84,191)(68,218,85,172)(69,271,86,153)(70,252,87,206)(71,233,88,187)(72,286,89,168)>;
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), (1,267,90,149)(2,248,91,202)(3,229,92,183)(4,282,93,164)(5,263,94,145)(6,244,95,198)(7,225,96,179)(8,278,97,160)(9,259,98,213)(10,240,99,194)(11,221,100,175)(12,274,101,156)(13,255,102,209)(14,236,103,190)(15,217,104,171)(16,270,105,152)(17,251,106,205)(18,232,107,186)(19,285,108,167)(20,266,109,148)(21,247,110,201)(22,228,111,182)(23,281,112,163)(24,262,113,216)(25,243,114,197)(26,224,115,178)(27,277,116,159)(28,258,117,212)(29,239,118,193)(30,220,119,174)(31,273,120,155)(32,254,121,208)(33,235,122,189)(34,288,123,170)(35,269,124,151)(36,250,125,204)(37,231,126,185)(38,284,127,166)(39,265,128,147)(40,246,129,200)(41,227,130,181)(42,280,131,162)(43,261,132,215)(44,242,133,196)(45,223,134,177)(46,276,135,158)(47,257,136,211)(48,238,137,192)(49,219,138,173)(50,272,139,154)(51,253,140,207)(52,234,141,188)(53,287,142,169)(54,268,143,150)(55,249,144,203)(56,230,73,184)(57,283,74,165)(58,264,75,146)(59,245,76,199)(60,226,77,180)(61,279,78,161)(62,260,79,214)(63,241,80,195)(64,222,81,176)(65,275,82,157)(66,256,83,210)(67,237,84,191)(68,218,85,172)(69,271,86,153)(70,252,87,206)(71,233,88,187)(72,286,89,168) );
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)], [(1,267,90,149),(2,248,91,202),(3,229,92,183),(4,282,93,164),(5,263,94,145),(6,244,95,198),(7,225,96,179),(8,278,97,160),(9,259,98,213),(10,240,99,194),(11,221,100,175),(12,274,101,156),(13,255,102,209),(14,236,103,190),(15,217,104,171),(16,270,105,152),(17,251,106,205),(18,232,107,186),(19,285,108,167),(20,266,109,148),(21,247,110,201),(22,228,111,182),(23,281,112,163),(24,262,113,216),(25,243,114,197),(26,224,115,178),(27,277,116,159),(28,258,117,212),(29,239,118,193),(30,220,119,174),(31,273,120,155),(32,254,121,208),(33,235,122,189),(34,288,123,170),(35,269,124,151),(36,250,125,204),(37,231,126,185),(38,284,127,166),(39,265,128,147),(40,246,129,200),(41,227,130,181),(42,280,131,162),(43,261,132,215),(44,242,133,196),(45,223,134,177),(46,276,135,158),(47,257,136,211),(48,238,137,192),(49,219,138,173),(50,272,139,154),(51,253,140,207),(52,234,141,188),(53,287,142,169),(54,268,143,150),(55,249,144,203),(56,230,73,184),(57,283,74,165),(58,264,75,146),(59,245,76,199),(60,226,77,180),(61,279,78,161),(62,260,79,214),(63,241,80,195),(64,222,81,176),(65,275,82,157),(66,256,83,210),(67,237,84,191),(68,218,85,172),(69,271,86,153),(70,252,87,206),(71,233,88,187),(72,286,89,168)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 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 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 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 | 1 | 1 | 1 | 1 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | - | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | Dic3 | D6 | M4(2) | D9 | C4×S3 | C4×S3 | Dic9 | D18 | C8⋊S3 | C4×D9 | C4×D9 | C8⋊D9 |
| kernel | C72⋊C4 | C2×C9⋊C8 | C4×Dic9 | C2×C72 | C9⋊C8 | C72 | C2×Dic9 | C2×C24 | C24 | C2×C12 | C18 | C2×C8 | C12 | C2×C6 | C8 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 1 | 2 | 1 | 4 | 3 | 2 | 2 | 6 | 3 | 8 | 6 | 6 | 24 |
Matrix representation of C72⋊C4 ►in GL3(𝔽73) generated by
| 27 | 0 | 0 |
| 0 | 39 | 20 |
| 0 | 53 | 19 |
| 27 | 0 | 0 |
| 0 | 63 | 41 |
| 0 | 51 | 10 |
G:=sub<GL(3,GF(73))| [27,0,0,0,39,53,0,20,19],[27,0,0,0,63,51,0,41,10] >;
C72⋊C4 in GAP, Magma, Sage, TeX
C_{72}\rtimes C_4 % in TeX
G:=Group("C72:C4"); // GroupNames label
G:=SmallGroup(288,23);
// by ID
G=gap.SmallGroup(288,23);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,253,64,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b|a^72=b^4=1,b*a*b^-1=a^53>;
// generators/relations
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