metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C76⋊1C4, C4⋊Dic19, C38.4D4, C2.1D76, C38.2Q8, C2.2Dic38, C22.5D38, C19⋊2(C4⋊C4), C38.8(C2×C4), (C2×C76).3C2, (C2×C4).3D19, (C2×C38).5C22, C2.4(C2×Dic19), (C2×Dic19).2C2, SmallGroup(304,12)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C76⋊C4
G = < a,b | a76=b4=1, bab-1=a-1 >
Smallest permutation representation of C76⋊C4
►Regular action on 304 points
Generators in S304
(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)
(1 148 185 247)(2 147 186 246)(3 146 187 245)(4 145 188 244)(5 144 189 243)(6 143 190 242)(7 142 191 241)(8 141 192 240)(9 140 193 239)(10 139 194 238)(11 138 195 237)(12 137 196 236)(13 136 197 235)(14 135 198 234)(15 134 199 233)(16 133 200 232)(17 132 201 231)(18 131 202 230)(19 130 203 229)(20 129 204 304)(21 128 205 303)(22 127 206 302)(23 126 207 301)(24 125 208 300)(25 124 209 299)(26 123 210 298)(27 122 211 297)(28 121 212 296)(29 120 213 295)(30 119 214 294)(31 118 215 293)(32 117 216 292)(33 116 217 291)(34 115 218 290)(35 114 219 289)(36 113 220 288)(37 112 221 287)(38 111 222 286)(39 110 223 285)(40 109 224 284)(41 108 225 283)(42 107 226 282)(43 106 227 281)(44 105 228 280)(45 104 153 279)(46 103 154 278)(47 102 155 277)(48 101 156 276)(49 100 157 275)(50 99 158 274)(51 98 159 273)(52 97 160 272)(53 96 161 271)(54 95 162 270)(55 94 163 269)(56 93 164 268)(57 92 165 267)(58 91 166 266)(59 90 167 265)(60 89 168 264)(61 88 169 263)(62 87 170 262)(63 86 171 261)(64 85 172 260)(65 84 173 259)(66 83 174 258)(67 82 175 257)(68 81 176 256)(69 80 177 255)(70 79 178 254)(71 78 179 253)(72 77 180 252)(73 152 181 251)(74 151 182 250)(75 150 183 249)(76 149 184 248)
G:=sub<Sym(304)| (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), (1,148,185,247)(2,147,186,246)(3,146,187,245)(4,145,188,244)(5,144,189,243)(6,143,190,242)(7,142,191,241)(8,141,192,240)(9,140,193,239)(10,139,194,238)(11,138,195,237)(12,137,196,236)(13,136,197,235)(14,135,198,234)(15,134,199,233)(16,133,200,232)(17,132,201,231)(18,131,202,230)(19,130,203,229)(20,129,204,304)(21,128,205,303)(22,127,206,302)(23,126,207,301)(24,125,208,300)(25,124,209,299)(26,123,210,298)(27,122,211,297)(28,121,212,296)(29,120,213,295)(30,119,214,294)(31,118,215,293)(32,117,216,292)(33,116,217,291)(34,115,218,290)(35,114,219,289)(36,113,220,288)(37,112,221,287)(38,111,222,286)(39,110,223,285)(40,109,224,284)(41,108,225,283)(42,107,226,282)(43,106,227,281)(44,105,228,280)(45,104,153,279)(46,103,154,278)(47,102,155,277)(48,101,156,276)(49,100,157,275)(50,99,158,274)(51,98,159,273)(52,97,160,272)(53,96,161,271)(54,95,162,270)(55,94,163,269)(56,93,164,268)(57,92,165,267)(58,91,166,266)(59,90,167,265)(60,89,168,264)(61,88,169,263)(62,87,170,262)(63,86,171,261)(64,85,172,260)(65,84,173,259)(66,83,174,258)(67,82,175,257)(68,81,176,256)(69,80,177,255)(70,79,178,254)(71,78,179,253)(72,77,180,252)(73,152,181,251)(74,151,182,250)(75,150,183,249)(76,149,184,248)>;
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)(229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304), (1,148,185,247)(2,147,186,246)(3,146,187,245)(4,145,188,244)(5,144,189,243)(6,143,190,242)(7,142,191,241)(8,141,192,240)(9,140,193,239)(10,139,194,238)(11,138,195,237)(12,137,196,236)(13,136,197,235)(14,135,198,234)(15,134,199,233)(16,133,200,232)(17,132,201,231)(18,131,202,230)(19,130,203,229)(20,129,204,304)(21,128,205,303)(22,127,206,302)(23,126,207,301)(24,125,208,300)(25,124,209,299)(26,123,210,298)(27,122,211,297)(28,121,212,296)(29,120,213,295)(30,119,214,294)(31,118,215,293)(32,117,216,292)(33,116,217,291)(34,115,218,290)(35,114,219,289)(36,113,220,288)(37,112,221,287)(38,111,222,286)(39,110,223,285)(40,109,224,284)(41,108,225,283)(42,107,226,282)(43,106,227,281)(44,105,228,280)(45,104,153,279)(46,103,154,278)(47,102,155,277)(48,101,156,276)(49,100,157,275)(50,99,158,274)(51,98,159,273)(52,97,160,272)(53,96,161,271)(54,95,162,270)(55,94,163,269)(56,93,164,268)(57,92,165,267)(58,91,166,266)(59,90,167,265)(60,89,168,264)(61,88,169,263)(62,87,170,262)(63,86,171,261)(64,85,172,260)(65,84,173,259)(66,83,174,258)(67,82,175,257)(68,81,176,256)(69,80,177,255)(70,79,178,254)(71,78,179,253)(72,77,180,252)(73,152,181,251)(74,151,182,250)(75,150,183,249)(76,149,184,248) );
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),(229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304)], [(1,148,185,247),(2,147,186,246),(3,146,187,245),(4,145,188,244),(5,144,189,243),(6,143,190,242),(7,142,191,241),(8,141,192,240),(9,140,193,239),(10,139,194,238),(11,138,195,237),(12,137,196,236),(13,136,197,235),(14,135,198,234),(15,134,199,233),(16,133,200,232),(17,132,201,231),(18,131,202,230),(19,130,203,229),(20,129,204,304),(21,128,205,303),(22,127,206,302),(23,126,207,301),(24,125,208,300),(25,124,209,299),(26,123,210,298),(27,122,211,297),(28,121,212,296),(29,120,213,295),(30,119,214,294),(31,118,215,293),(32,117,216,292),(33,116,217,291),(34,115,218,290),(35,114,219,289),(36,113,220,288),(37,112,221,287),(38,111,222,286),(39,110,223,285),(40,109,224,284),(41,108,225,283),(42,107,226,282),(43,106,227,281),(44,105,228,280),(45,104,153,279),(46,103,154,278),(47,102,155,277),(48,101,156,276),(49,100,157,275),(50,99,158,274),(51,98,159,273),(52,97,160,272),(53,96,161,271),(54,95,162,270),(55,94,163,269),(56,93,164,268),(57,92,165,267),(58,91,166,266),(59,90,167,265),(60,89,168,264),(61,88,169,263),(62,87,170,262),(63,86,171,261),(64,85,172,260),(65,84,173,259),(66,83,174,258),(67,82,175,257),(68,81,176,256),(69,80,177,255),(70,79,178,254),(71,78,179,253),(72,77,180,252),(73,152,181,251),(74,151,182,250),(75,150,183,249),(76,149,184,248)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 19A | ··· | 19I | 38A | ··· | 38AA | 76A | ··· | 76AJ |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 19 | ··· | 19 | 38 | ··· | 38 | 76 | ··· | 76 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 38 | 38 | 38 | 38 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | - | + | |
| image | C1 | C2 | C2 | C4 | D4 | Q8 | D19 | Dic19 | D38 | Dic38 | D76 |
| kernel | C76⋊C4 | C2×Dic19 | C2×C76 | C76 | C38 | C38 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 9 | 18 | 9 | 18 | 18 |
Matrix representation of C76⋊C4 ►in GL3(𝔽229) generated by
| 1 | 0 | 0 |
| 0 | 87 | 148 |
| 0 | 16 | 172 |
| 107 | 0 | 0 |
| 0 | 206 | 208 |
| 0 | 167 | 23 |
G:=sub<GL(3,GF(229))| [1,0,0,0,87,16,0,148,172],[107,0,0,0,206,167,0,208,23] >;
C76⋊C4 in GAP, Magma, Sage, TeX
C_{76}\rtimes C_4 % in TeX
G:=Group("C76:C4"); // GroupNames label
G:=SmallGroup(304,12);
// by ID
G=gap.SmallGroup(304,12);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-19,20,101,46,7204]);
// Polycyclic
G:=Group<a,b|a^76=b^4=1,b*a*b^-1=a^-1>;
// generators/relations
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