direct product, abelian, monomial, 2-elementary
Aliases: C4×C76, SmallGroup(304,19)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C4×C76 |
| C1 — C4×C76 |
| C1 — C4×C76 |
Generators and relations for C4×C76
G = < a,b | a4=b76=1, ab=ba >
Smallest permutation representation of C4×C76
►Regular action on 304 points
Generators in S304
(1 180 145 286)(2 181 146 287)(3 182 147 288)(4 183 148 289)(5 184 149 290)(6 185 150 291)(7 186 151 292)(8 187 152 293)(9 188 77 294)(10 189 78 295)(11 190 79 296)(12 191 80 297)(13 192 81 298)(14 193 82 299)(15 194 83 300)(16 195 84 301)(17 196 85 302)(18 197 86 303)(19 198 87 304)(20 199 88 229)(21 200 89 230)(22 201 90 231)(23 202 91 232)(24 203 92 233)(25 204 93 234)(26 205 94 235)(27 206 95 236)(28 207 96 237)(29 208 97 238)(30 209 98 239)(31 210 99 240)(32 211 100 241)(33 212 101 242)(34 213 102 243)(35 214 103 244)(36 215 104 245)(37 216 105 246)(38 217 106 247)(39 218 107 248)(40 219 108 249)(41 220 109 250)(42 221 110 251)(43 222 111 252)(44 223 112 253)(45 224 113 254)(46 225 114 255)(47 226 115 256)(48 227 116 257)(49 228 117 258)(50 153 118 259)(51 154 119 260)(52 155 120 261)(53 156 121 262)(54 157 122 263)(55 158 123 264)(56 159 124 265)(57 160 125 266)(58 161 126 267)(59 162 127 268)(60 163 128 269)(61 164 129 270)(62 165 130 271)(63 166 131 272)(64 167 132 273)(65 168 133 274)(66 169 134 275)(67 170 135 276)(68 171 136 277)(69 172 137 278)(70 173 138 279)(71 174 139 280)(72 175 140 281)(73 176 141 282)(74 177 142 283)(75 178 143 284)(76 179 144 285)
(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)(229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304)
G:=sub<Sym(304)| (1,180,145,286)(2,181,146,287)(3,182,147,288)(4,183,148,289)(5,184,149,290)(6,185,150,291)(7,186,151,292)(8,187,152,293)(9,188,77,294)(10,189,78,295)(11,190,79,296)(12,191,80,297)(13,192,81,298)(14,193,82,299)(15,194,83,300)(16,195,84,301)(17,196,85,302)(18,197,86,303)(19,198,87,304)(20,199,88,229)(21,200,89,230)(22,201,90,231)(23,202,91,232)(24,203,92,233)(25,204,93,234)(26,205,94,235)(27,206,95,236)(28,207,96,237)(29,208,97,238)(30,209,98,239)(31,210,99,240)(32,211,100,241)(33,212,101,242)(34,213,102,243)(35,214,103,244)(36,215,104,245)(37,216,105,246)(38,217,106,247)(39,218,107,248)(40,219,108,249)(41,220,109,250)(42,221,110,251)(43,222,111,252)(44,223,112,253)(45,224,113,254)(46,225,114,255)(47,226,115,256)(48,227,116,257)(49,228,117,258)(50,153,118,259)(51,154,119,260)(52,155,120,261)(53,156,121,262)(54,157,122,263)(55,158,123,264)(56,159,124,265)(57,160,125,266)(58,161,126,267)(59,162,127,268)(60,163,128,269)(61,164,129,270)(62,165,130,271)(63,166,131,272)(64,167,132,273)(65,168,133,274)(66,169,134,275)(67,170,135,276)(68,171,136,277)(69,172,137,278)(70,173,138,279)(71,174,139,280)(72,175,140,281)(73,176,141,282)(74,177,142,283)(75,178,143,284)(76,179,144,285), (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)(229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304)>;
G:=Group( (1,180,145,286)(2,181,146,287)(3,182,147,288)(4,183,148,289)(5,184,149,290)(6,185,150,291)(7,186,151,292)(8,187,152,293)(9,188,77,294)(10,189,78,295)(11,190,79,296)(12,191,80,297)(13,192,81,298)(14,193,82,299)(15,194,83,300)(16,195,84,301)(17,196,85,302)(18,197,86,303)(19,198,87,304)(20,199,88,229)(21,200,89,230)(22,201,90,231)(23,202,91,232)(24,203,92,233)(25,204,93,234)(26,205,94,235)(27,206,95,236)(28,207,96,237)(29,208,97,238)(30,209,98,239)(31,210,99,240)(32,211,100,241)(33,212,101,242)(34,213,102,243)(35,214,103,244)(36,215,104,245)(37,216,105,246)(38,217,106,247)(39,218,107,248)(40,219,108,249)(41,220,109,250)(42,221,110,251)(43,222,111,252)(44,223,112,253)(45,224,113,254)(46,225,114,255)(47,226,115,256)(48,227,116,257)(49,228,117,258)(50,153,118,259)(51,154,119,260)(52,155,120,261)(53,156,121,262)(54,157,122,263)(55,158,123,264)(56,159,124,265)(57,160,125,266)(58,161,126,267)(59,162,127,268)(60,163,128,269)(61,164,129,270)(62,165,130,271)(63,166,131,272)(64,167,132,273)(65,168,133,274)(66,169,134,275)(67,170,135,276)(68,171,136,277)(69,172,137,278)(70,173,138,279)(71,174,139,280)(72,175,140,281)(73,176,141,282)(74,177,142,283)(75,178,143,284)(76,179,144,285), (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)(229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304) );
G=PermutationGroup([[(1,180,145,286),(2,181,146,287),(3,182,147,288),(4,183,148,289),(5,184,149,290),(6,185,150,291),(7,186,151,292),(8,187,152,293),(9,188,77,294),(10,189,78,295),(11,190,79,296),(12,191,80,297),(13,192,81,298),(14,193,82,299),(15,194,83,300),(16,195,84,301),(17,196,85,302),(18,197,86,303),(19,198,87,304),(20,199,88,229),(21,200,89,230),(22,201,90,231),(23,202,91,232),(24,203,92,233),(25,204,93,234),(26,205,94,235),(27,206,95,236),(28,207,96,237),(29,208,97,238),(30,209,98,239),(31,210,99,240),(32,211,100,241),(33,212,101,242),(34,213,102,243),(35,214,103,244),(36,215,104,245),(37,216,105,246),(38,217,106,247),(39,218,107,248),(40,219,108,249),(41,220,109,250),(42,221,110,251),(43,222,111,252),(44,223,112,253),(45,224,113,254),(46,225,114,255),(47,226,115,256),(48,227,116,257),(49,228,117,258),(50,153,118,259),(51,154,119,260),(52,155,120,261),(53,156,121,262),(54,157,122,263),(55,158,123,264),(56,159,124,265),(57,160,125,266),(58,161,126,267),(59,162,127,268),(60,163,128,269),(61,164,129,270),(62,165,130,271),(63,166,131,272),(64,167,132,273),(65,168,133,274),(66,169,134,275),(67,170,135,276),(68,171,136,277),(69,172,137,278),(70,173,138,279),(71,174,139,280),(72,175,140,281),(73,176,141,282),(74,177,142,283),(75,178,143,284),(76,179,144,285)], [(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),(229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304)]])
304 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4L | 19A | ··· | 19R | 38A | ··· | 38BB | 76A | ··· | 76HH |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 19 | ··· | 19 | 38 | ··· | 38 | 76 | ··· | 76 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
304 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||
| image | C1 | C2 | C4 | C19 | C38 | C76 |
| kernel | C4×C76 | C2×C76 | C76 | C42 | C2×C4 | C4 |
| # reps | 1 | 3 | 12 | 18 | 54 | 216 |
Matrix representation of C4×C76 ►in GL2(𝔽229) generated by
| 1 | 0 |
| 0 | 107 |
| 93 | 0 |
| 0 | 141 |
G:=sub<GL(2,GF(229))| [1,0,0,107],[93,0,0,141] >;
C4×C76 in GAP, Magma, Sage, TeX
C_4\times C_{76} % in TeX
G:=Group("C4xC76"); // GroupNames label
G:=SmallGroup(304,19);
// by ID
G=gap.SmallGroup(304,19);
# by ID
G:=PCGroup([5,-2,-2,-19,-2,-2,380,766]);
// Polycyclic
G:=Group<a,b|a^4=b^76=1,a*b=b*a>;
// generators/relations
Export