metacyclic, supersoluble, monomial, Z-group
Aliases: C7⋊C36, Dic7⋊C9, C14.C18, C21.C12, C6.2F7, C42.2C6, C7⋊C9⋊C4, C3.(C7⋊C12), C2.(C7⋊C18), (C3×Dic7).C3, (C2×C7⋊C9).C2, SmallGroup(252,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C7⋊C36 |
Generators and relations for C7⋊C36
G = < a,b | a7=b36=1, bab-1=a5 >
Smallest permutation representation of C7⋊C36
►Regular action on 252 points
Generators in S252
(1 85 136 46 160 184 239)(2 185 47 86 240 161 137)(3 162 87 186 138 241 48)(4 242 187 163 49 139 88)(5 140 164 243 89 50 188)(6 51 244 141 189 90 165)(7 91 142 52 166 190 245)(8 191 53 92 246 167 143)(9 168 93 192 144 247 54)(10 248 193 169 55 109 94)(11 110 170 249 95 56 194)(12 57 250 111 195 96 171)(13 97 112 58 172 196 251)(14 197 59 98 252 173 113)(15 174 99 198 114 217 60)(16 218 199 175 61 115 100)(17 116 176 219 101 62 200)(18 63 220 117 201 102 177)(19 103 118 64 178 202 221)(20 203 65 104 222 179 119)(21 180 105 204 120 223 66)(22 224 205 145 67 121 106)(23 122 146 225 107 68 206)(24 69 226 123 207 108 147)(25 73 124 70 148 208 227)(26 209 71 74 228 149 125)(27 150 75 210 126 229 72)(28 230 211 151 37 127 76)(29 128 152 231 77 38 212)(30 39 232 129 213 78 153)(31 79 130 40 154 214 233)(32 215 41 80 234 155 131)(33 156 81 216 132 235 42)(34 236 181 157 43 133 82)(35 134 158 237 83 44 182)(36 45 238 135 183 84 159)
(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)
G:=sub<Sym(252)| (1,85,136,46,160,184,239)(2,185,47,86,240,161,137)(3,162,87,186,138,241,48)(4,242,187,163,49,139,88)(5,140,164,243,89,50,188)(6,51,244,141,189,90,165)(7,91,142,52,166,190,245)(8,191,53,92,246,167,143)(9,168,93,192,144,247,54)(10,248,193,169,55,109,94)(11,110,170,249,95,56,194)(12,57,250,111,195,96,171)(13,97,112,58,172,196,251)(14,197,59,98,252,173,113)(15,174,99,198,114,217,60)(16,218,199,175,61,115,100)(17,116,176,219,101,62,200)(18,63,220,117,201,102,177)(19,103,118,64,178,202,221)(20,203,65,104,222,179,119)(21,180,105,204,120,223,66)(22,224,205,145,67,121,106)(23,122,146,225,107,68,206)(24,69,226,123,207,108,147)(25,73,124,70,148,208,227)(26,209,71,74,228,149,125)(27,150,75,210,126,229,72)(28,230,211,151,37,127,76)(29,128,152,231,77,38,212)(30,39,232,129,213,78,153)(31,79,130,40,154,214,233)(32,215,41,80,234,155,131)(33,156,81,216,132,235,42)(34,236,181,157,43,133,82)(35,134,158,237,83,44,182)(36,45,238,135,183,84,159), (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)>;
G:=Group( (1,85,136,46,160,184,239)(2,185,47,86,240,161,137)(3,162,87,186,138,241,48)(4,242,187,163,49,139,88)(5,140,164,243,89,50,188)(6,51,244,141,189,90,165)(7,91,142,52,166,190,245)(8,191,53,92,246,167,143)(9,168,93,192,144,247,54)(10,248,193,169,55,109,94)(11,110,170,249,95,56,194)(12,57,250,111,195,96,171)(13,97,112,58,172,196,251)(14,197,59,98,252,173,113)(15,174,99,198,114,217,60)(16,218,199,175,61,115,100)(17,116,176,219,101,62,200)(18,63,220,117,201,102,177)(19,103,118,64,178,202,221)(20,203,65,104,222,179,119)(21,180,105,204,120,223,66)(22,224,205,145,67,121,106)(23,122,146,225,107,68,206)(24,69,226,123,207,108,147)(25,73,124,70,148,208,227)(26,209,71,74,228,149,125)(27,150,75,210,126,229,72)(28,230,211,151,37,127,76)(29,128,152,231,77,38,212)(30,39,232,129,213,78,153)(31,79,130,40,154,214,233)(32,215,41,80,234,155,131)(33,156,81,216,132,235,42)(34,236,181,157,43,133,82)(35,134,158,237,83,44,182)(36,45,238,135,183,84,159), (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) );
G=PermutationGroup([[(1,85,136,46,160,184,239),(2,185,47,86,240,161,137),(3,162,87,186,138,241,48),(4,242,187,163,49,139,88),(5,140,164,243,89,50,188),(6,51,244,141,189,90,165),(7,91,142,52,166,190,245),(8,191,53,92,246,167,143),(9,168,93,192,144,247,54),(10,248,193,169,55,109,94),(11,110,170,249,95,56,194),(12,57,250,111,195,96,171),(13,97,112,58,172,196,251),(14,197,59,98,252,173,113),(15,174,99,198,114,217,60),(16,218,199,175,61,115,100),(17,116,176,219,101,62,200),(18,63,220,117,201,102,177),(19,103,118,64,178,202,221),(20,203,65,104,222,179,119),(21,180,105,204,120,223,66),(22,224,205,145,67,121,106),(23,122,146,225,107,68,206),(24,69,226,123,207,108,147),(25,73,124,70,148,208,227),(26,209,71,74,228,149,125),(27,150,75,210,126,229,72),(28,230,211,151,37,127,76),(29,128,152,231,77,38,212),(30,39,232,129,213,78,153),(31,79,130,40,154,214,233),(32,215,41,80,234,155,131),(33,156,81,216,132,235,42),(34,236,181,157,43,133,82),(35,134,158,237,83,44,182),(36,45,238,135,183,84,159)], [(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)]])
42 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 7 | 9A | ··· | 9F | 12A | 12B | 12C | 12D | 14 | 18A | ··· | 18F | 21A | 21B | 36A | ··· | 36L | 42A | 42B |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 7 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 14 | 18 | ··· | 18 | 21 | 21 | 36 | ··· | 36 | 42 | 42 |
| size | 1 | 1 | 1 | 1 | 7 | 7 | 1 | 1 | 6 | 7 | ··· | 7 | 7 | 7 | 7 | 7 | 6 | 7 | ··· | 7 | 6 | 6 | 7 | ··· | 7 | 6 | 6 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 |
| type | + | + | + | - | |||||||||
| image | C1 | C2 | C3 | C4 | C6 | C9 | C12 | C18 | C36 | F7 | C7⋊C12 | C7⋊C18 | C7⋊C36 |
| kernel | C7⋊C36 | C2×C7⋊C9 | C3×Dic7 | C7⋊C9 | C42 | Dic7 | C21 | C14 | C7 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 6 | 4 | 6 | 12 | 1 | 1 | 2 | 2 |
Matrix representation of C7⋊C36 ►in GL7(𝔽757)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 756 | 1 | 0 | 0 | 0 | 0 |
| 0 | 756 | 0 | 1 | 0 | 0 | 0 |
| 0 | 756 | 0 | 0 | 1 | 0 | 0 |
| 0 | 756 | 0 | 0 | 0 | 1 | 0 |
| 0 | 756 | 0 | 0 | 0 | 0 | 1 |
| 0 | 756 | 0 | 0 | 0 | 0 | 0 |
| 87 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 93 | 353 | 619 | 435 | 629 | 504 |
| 0 | 712 | 31 | 491 | 182 | 267 | 597 |
| 0 | 584 | 535 | 129 | 275 | 620 | 459 |
| 0 | 222 | 628 | 482 | 137 | 298 | 331 |
| 0 | 575 | 490 | 160 | 9 | 45 | 726 |
| 0 | 253 | 362 | 664 | 404 | 138 | 322 |
G:=sub<GL(7,GF(757))| [1,0,0,0,0,0,0,0,756,756,756,756,756,756,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0],[87,0,0,0,0,0,0,0,93,712,584,222,575,253,0,353,31,535,628,490,362,0,619,491,129,482,160,664,0,435,182,275,137,9,404,0,629,267,620,298,45,138,0,504,597,459,331,726,322] >;
C7⋊C36 in GAP, Magma, Sage, TeX
C_7\rtimes C_{36} % in TeX
G:=Group("C7:C36"); // GroupNames label
G:=SmallGroup(252,1);
// by ID
G=gap.SmallGroup(252,1);
# by ID
G:=PCGroup([5,-2,-3,-2,-3,-7,30,66,5404,1809]);
// Polycyclic
G:=Group<a,b|a^7=b^36=1,b*a*b^-1=a^5>;
// generators/relations
Export