direct product, abelian, monomial, 3-elementary
Aliases: C3×C96, SmallGroup(288,66)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C96 |
| C1 — C3×C96 |
| C1 — C3×C96 |
Generators and relations for C3×C96
G = < a,b | a3=b96=1, ab=ba >
Smallest permutation representation of C3×C96
►Regular action on 288 points
Generators in S288
(1 102 266)(2 103 267)(3 104 268)(4 105 269)(5 106 270)(6 107 271)(7 108 272)(8 109 273)(9 110 274)(10 111 275)(11 112 276)(12 113 277)(13 114 278)(14 115 279)(15 116 280)(16 117 281)(17 118 282)(18 119 283)(19 120 284)(20 121 285)(21 122 286)(22 123 287)(23 124 288)(24 125 193)(25 126 194)(26 127 195)(27 128 196)(28 129 197)(29 130 198)(30 131 199)(31 132 200)(32 133 201)(33 134 202)(34 135 203)(35 136 204)(36 137 205)(37 138 206)(38 139 207)(39 140 208)(40 141 209)(41 142 210)(42 143 211)(43 144 212)(44 145 213)(45 146 214)(46 147 215)(47 148 216)(48 149 217)(49 150 218)(50 151 219)(51 152 220)(52 153 221)(53 154 222)(54 155 223)(55 156 224)(56 157 225)(57 158 226)(58 159 227)(59 160 228)(60 161 229)(61 162 230)(62 163 231)(63 164 232)(64 165 233)(65 166 234)(66 167 235)(67 168 236)(68 169 237)(69 170 238)(70 171 239)(71 172 240)(72 173 241)(73 174 242)(74 175 243)(75 176 244)(76 177 245)(77 178 246)(78 179 247)(79 180 248)(80 181 249)(81 182 250)(82 183 251)(83 184 252)(84 185 253)(85 186 254)(86 187 255)(87 188 256)(88 189 257)(89 190 258)(90 191 259)(91 192 260)(92 97 261)(93 98 262)(94 99 263)(95 100 264)(96 101 265)
(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)
G:=sub<Sym(288)| (1,102,266)(2,103,267)(3,104,268)(4,105,269)(5,106,270)(6,107,271)(7,108,272)(8,109,273)(9,110,274)(10,111,275)(11,112,276)(12,113,277)(13,114,278)(14,115,279)(15,116,280)(16,117,281)(17,118,282)(18,119,283)(19,120,284)(20,121,285)(21,122,286)(22,123,287)(23,124,288)(24,125,193)(25,126,194)(26,127,195)(27,128,196)(28,129,197)(29,130,198)(30,131,199)(31,132,200)(32,133,201)(33,134,202)(34,135,203)(35,136,204)(36,137,205)(37,138,206)(38,139,207)(39,140,208)(40,141,209)(41,142,210)(42,143,211)(43,144,212)(44,145,213)(45,146,214)(46,147,215)(47,148,216)(48,149,217)(49,150,218)(50,151,219)(51,152,220)(52,153,221)(53,154,222)(54,155,223)(55,156,224)(56,157,225)(57,158,226)(58,159,227)(59,160,228)(60,161,229)(61,162,230)(62,163,231)(63,164,232)(64,165,233)(65,166,234)(66,167,235)(67,168,236)(68,169,237)(69,170,238)(70,171,239)(71,172,240)(72,173,241)(73,174,242)(74,175,243)(75,176,244)(76,177,245)(77,178,246)(78,179,247)(79,180,248)(80,181,249)(81,182,250)(82,183,251)(83,184,252)(84,185,253)(85,186,254)(86,187,255)(87,188,256)(88,189,257)(89,190,258)(90,191,259)(91,192,260)(92,97,261)(93,98,262)(94,99,263)(95,100,264)(96,101,265), (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)>;
G:=Group( (1,102,266)(2,103,267)(3,104,268)(4,105,269)(5,106,270)(6,107,271)(7,108,272)(8,109,273)(9,110,274)(10,111,275)(11,112,276)(12,113,277)(13,114,278)(14,115,279)(15,116,280)(16,117,281)(17,118,282)(18,119,283)(19,120,284)(20,121,285)(21,122,286)(22,123,287)(23,124,288)(24,125,193)(25,126,194)(26,127,195)(27,128,196)(28,129,197)(29,130,198)(30,131,199)(31,132,200)(32,133,201)(33,134,202)(34,135,203)(35,136,204)(36,137,205)(37,138,206)(38,139,207)(39,140,208)(40,141,209)(41,142,210)(42,143,211)(43,144,212)(44,145,213)(45,146,214)(46,147,215)(47,148,216)(48,149,217)(49,150,218)(50,151,219)(51,152,220)(52,153,221)(53,154,222)(54,155,223)(55,156,224)(56,157,225)(57,158,226)(58,159,227)(59,160,228)(60,161,229)(61,162,230)(62,163,231)(63,164,232)(64,165,233)(65,166,234)(66,167,235)(67,168,236)(68,169,237)(69,170,238)(70,171,239)(71,172,240)(72,173,241)(73,174,242)(74,175,243)(75,176,244)(76,177,245)(77,178,246)(78,179,247)(79,180,248)(80,181,249)(81,182,250)(82,183,251)(83,184,252)(84,185,253)(85,186,254)(86,187,255)(87,188,256)(88,189,257)(89,190,258)(90,191,259)(91,192,260)(92,97,261)(93,98,262)(94,99,263)(95,100,264)(96,101,265), (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) );
G=PermutationGroup([[(1,102,266),(2,103,267),(3,104,268),(4,105,269),(5,106,270),(6,107,271),(7,108,272),(8,109,273),(9,110,274),(10,111,275),(11,112,276),(12,113,277),(13,114,278),(14,115,279),(15,116,280),(16,117,281),(17,118,282),(18,119,283),(19,120,284),(20,121,285),(21,122,286),(22,123,287),(23,124,288),(24,125,193),(25,126,194),(26,127,195),(27,128,196),(28,129,197),(29,130,198),(30,131,199),(31,132,200),(32,133,201),(33,134,202),(34,135,203),(35,136,204),(36,137,205),(37,138,206),(38,139,207),(39,140,208),(40,141,209),(41,142,210),(42,143,211),(43,144,212),(44,145,213),(45,146,214),(46,147,215),(47,148,216),(48,149,217),(49,150,218),(50,151,219),(51,152,220),(52,153,221),(53,154,222),(54,155,223),(55,156,224),(56,157,225),(57,158,226),(58,159,227),(59,160,228),(60,161,229),(61,162,230),(62,163,231),(63,164,232),(64,165,233),(65,166,234),(66,167,235),(67,168,236),(68,169,237),(69,170,238),(70,171,239),(71,172,240),(72,173,241),(73,174,242),(74,175,243),(75,176,244),(76,177,245),(77,178,246),(78,179,247),(79,180,248),(80,181,249),(81,182,250),(82,183,251),(83,184,252),(84,185,253),(85,186,254),(86,187,255),(87,188,256),(88,189,257),(89,190,258),(90,191,259),(91,192,260),(92,97,261),(93,98,262),(94,99,263),(95,100,264),(96,101,265)], [(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)]])
288 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 4A | 4B | 6A | ··· | 6H | 8A | 8B | 8C | 8D | 12A | ··· | 12P | 16A | ··· | 16H | 24A | ··· | 24AF | 32A | ··· | 32P | 48A | ··· | 48BL | 96A | ··· | 96DX |
| order | 1 | 2 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 16 | ··· | 16 | 24 | ··· | 24 | 32 | ··· | 32 | 48 | ··· | 48 | 96 | ··· | 96 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
288 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C12 | C16 | C24 | C32 | C48 | C96 |
| kernel | C3×C96 | C3×C48 | C96 | C3×C24 | C48 | C3×C12 | C24 | C3×C6 | C12 | C32 | C6 | C3 |
| # reps | 1 | 1 | 8 | 2 | 8 | 4 | 16 | 8 | 32 | 16 | 64 | 128 |
Matrix representation of C3×C96 ►in GL2(𝔽97) generated by
| 35 | 0 |
| 0 | 35 |
| 82 | 0 |
| 0 | 29 |
G:=sub<GL(2,GF(97))| [35,0,0,35],[82,0,0,29] >;
C3×C96 in GAP, Magma, Sage, TeX
C_3\times C_{96} % in TeX
G:=Group("C3xC96"); // GroupNames label
G:=SmallGroup(288,66);
// by ID
G=gap.SmallGroup(288,66);
# by ID
G:=PCGroup([7,-2,-3,-3,-2,-2,-2,-2,126,80,102,124]);
// Polycyclic
G:=Group<a,b|a^3=b^96=1,a*b=b*a>;
// generators/relations
Export