metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C37⋊2C8, C74.2C4, C4.2D37, C2.Dic37, C148.2C2, SmallGroup(296,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C37 — C37⋊2C8 |
Generators and relations for C37⋊2C8
G = < a,b | a37=b8=1, bab-1=a-1 >
Smallest permutation representation of C37⋊2C8
►Regular action on 296 points
Generators in S296
(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)
(1 260 129 186 56 223 94 149)(2 296 130 222 57 259 95 185)(3 295 131 221 58 258 96 184)(4 294 132 220 59 257 97 183)(5 293 133 219 60 256 98 182)(6 292 134 218 61 255 99 181)(7 291 135 217 62 254 100 180)(8 290 136 216 63 253 101 179)(9 289 137 215 64 252 102 178)(10 288 138 214 65 251 103 177)(11 287 139 213 66 250 104 176)(12 286 140 212 67 249 105 175)(13 285 141 211 68 248 106 174)(14 284 142 210 69 247 107 173)(15 283 143 209 70 246 108 172)(16 282 144 208 71 245 109 171)(17 281 145 207 72 244 110 170)(18 280 146 206 73 243 111 169)(19 279 147 205 74 242 75 168)(20 278 148 204 38 241 76 167)(21 277 112 203 39 240 77 166)(22 276 113 202 40 239 78 165)(23 275 114 201 41 238 79 164)(24 274 115 200 42 237 80 163)(25 273 116 199 43 236 81 162)(26 272 117 198 44 235 82 161)(27 271 118 197 45 234 83 160)(28 270 119 196 46 233 84 159)(29 269 120 195 47 232 85 158)(30 268 121 194 48 231 86 157)(31 267 122 193 49 230 87 156)(32 266 123 192 50 229 88 155)(33 265 124 191 51 228 89 154)(34 264 125 190 52 227 90 153)(35 263 126 189 53 226 91 152)(36 262 127 188 54 225 92 151)(37 261 128 187 55 224 93 150)
G:=sub<Sym(296)| (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), (1,260,129,186,56,223,94,149)(2,296,130,222,57,259,95,185)(3,295,131,221,58,258,96,184)(4,294,132,220,59,257,97,183)(5,293,133,219,60,256,98,182)(6,292,134,218,61,255,99,181)(7,291,135,217,62,254,100,180)(8,290,136,216,63,253,101,179)(9,289,137,215,64,252,102,178)(10,288,138,214,65,251,103,177)(11,287,139,213,66,250,104,176)(12,286,140,212,67,249,105,175)(13,285,141,211,68,248,106,174)(14,284,142,210,69,247,107,173)(15,283,143,209,70,246,108,172)(16,282,144,208,71,245,109,171)(17,281,145,207,72,244,110,170)(18,280,146,206,73,243,111,169)(19,279,147,205,74,242,75,168)(20,278,148,204,38,241,76,167)(21,277,112,203,39,240,77,166)(22,276,113,202,40,239,78,165)(23,275,114,201,41,238,79,164)(24,274,115,200,42,237,80,163)(25,273,116,199,43,236,81,162)(26,272,117,198,44,235,82,161)(27,271,118,197,45,234,83,160)(28,270,119,196,46,233,84,159)(29,269,120,195,47,232,85,158)(30,268,121,194,48,231,86,157)(31,267,122,193,49,230,87,156)(32,266,123,192,50,229,88,155)(33,265,124,191,51,228,89,154)(34,264,125,190,52,227,90,153)(35,263,126,189,53,226,91,152)(36,262,127,188,54,225,92,151)(37,261,128,187,55,224,93,150)>;
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), (1,260,129,186,56,223,94,149)(2,296,130,222,57,259,95,185)(3,295,131,221,58,258,96,184)(4,294,132,220,59,257,97,183)(5,293,133,219,60,256,98,182)(6,292,134,218,61,255,99,181)(7,291,135,217,62,254,100,180)(8,290,136,216,63,253,101,179)(9,289,137,215,64,252,102,178)(10,288,138,214,65,251,103,177)(11,287,139,213,66,250,104,176)(12,286,140,212,67,249,105,175)(13,285,141,211,68,248,106,174)(14,284,142,210,69,247,107,173)(15,283,143,209,70,246,108,172)(16,282,144,208,71,245,109,171)(17,281,145,207,72,244,110,170)(18,280,146,206,73,243,111,169)(19,279,147,205,74,242,75,168)(20,278,148,204,38,241,76,167)(21,277,112,203,39,240,77,166)(22,276,113,202,40,239,78,165)(23,275,114,201,41,238,79,164)(24,274,115,200,42,237,80,163)(25,273,116,199,43,236,81,162)(26,272,117,198,44,235,82,161)(27,271,118,197,45,234,83,160)(28,270,119,196,46,233,84,159)(29,269,120,195,47,232,85,158)(30,268,121,194,48,231,86,157)(31,267,122,193,49,230,87,156)(32,266,123,192,50,229,88,155)(33,265,124,191,51,228,89,154)(34,264,125,190,52,227,90,153)(35,263,126,189,53,226,91,152)(36,262,127,188,54,225,92,151)(37,261,128,187,55,224,93,150) );
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)], [(1,260,129,186,56,223,94,149),(2,296,130,222,57,259,95,185),(3,295,131,221,58,258,96,184),(4,294,132,220,59,257,97,183),(5,293,133,219,60,256,98,182),(6,292,134,218,61,255,99,181),(7,291,135,217,62,254,100,180),(8,290,136,216,63,253,101,179),(9,289,137,215,64,252,102,178),(10,288,138,214,65,251,103,177),(11,287,139,213,66,250,104,176),(12,286,140,212,67,249,105,175),(13,285,141,211,68,248,106,174),(14,284,142,210,69,247,107,173),(15,283,143,209,70,246,108,172),(16,282,144,208,71,245,109,171),(17,281,145,207,72,244,110,170),(18,280,146,206,73,243,111,169),(19,279,147,205,74,242,75,168),(20,278,148,204,38,241,76,167),(21,277,112,203,39,240,77,166),(22,276,113,202,40,239,78,165),(23,275,114,201,41,238,79,164),(24,274,115,200,42,237,80,163),(25,273,116,199,43,236,81,162),(26,272,117,198,44,235,82,161),(27,271,118,197,45,234,83,160),(28,270,119,196,46,233,84,159),(29,269,120,195,47,232,85,158),(30,268,121,194,48,231,86,157),(31,267,122,193,49,230,87,156),(32,266,123,192,50,229,88,155),(33,265,124,191,51,228,89,154),(34,264,125,190,52,227,90,153),(35,263,126,189,53,226,91,152),(36,262,127,188,54,225,92,151),(37,261,128,187,55,224,93,150)]])
80 conjugacy classes
| class | 1 | 2 | 4A | 4B | 8A | 8B | 8C | 8D | 37A | ··· | 37R | 74A | ··· | 74R | 148A | ··· | 148AJ |
| order | 1 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 37 | ··· | 37 | 74 | ··· | 74 | 148 | ··· | 148 |
| size | 1 | 1 | 1 | 1 | 37 | 37 | 37 | 37 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | - | |||
| image | C1 | C2 | C4 | C8 | D37 | Dic37 | C37⋊2C8 |
| kernel | C37⋊2C8 | C148 | C74 | C37 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 18 | 18 | 36 |
Matrix representation of C37⋊2C8 ►in GL2(𝔽593) generated by
| 0 | 1 |
| 592 | 561 |
| 352 | 426 |
| 429 | 241 |
G:=sub<GL(2,GF(593))| [0,592,1,561],[352,429,426,241] >;
C37⋊2C8 in GAP, Magma, Sage, TeX
C_{37}\rtimes_2C_8 % in TeX
G:=Group("C37:2C8"); // GroupNames label
G:=SmallGroup(296,1);
// by ID
G=gap.SmallGroup(296,1);
# by ID
G:=PCGroup([4,-2,-2,-2,-37,8,21,4611]);
// Polycyclic
G:=Group<a,b|a^37=b^8=1,b*a*b^-1=a^-1>;
// generators/relations
Export