metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C7⋊C27, C63.C3, C21.C9, C3.(C7⋊C9), C9.(C7⋊C3), SmallGroup(189,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C7⋊C27 |
Generators and relations for C7⋊C27
G = < a,b | a7=b27=1, bab-1=a4 >
Smallest permutation representation of C7⋊C27
►Regular action on 189 points
Generators in S189
(1 48 114 158 102 68 175)(2 103 49 69 115 176 159)(3 116 104 177 50 160 70)(4 51 117 161 105 71 178)(5 106 52 72 118 179 162)(6 119 107 180 53 136 73)(7 54 120 137 108 74 181)(8 82 28 75 121 182 138)(9 122 83 183 29 139 76)(10 30 123 140 84 77 184)(11 85 31 78 124 185 141)(12 125 86 186 32 142 79)(13 33 126 143 87 80 187)(14 88 34 81 127 188 144)(15 128 89 189 35 145 55)(16 36 129 146 90 56 163)(17 91 37 57 130 164 147)(18 131 92 165 38 148 58)(19 39 132 149 93 59 166)(20 94 40 60 133 167 150)(21 134 95 168 41 151 61)(22 42 135 152 96 62 169)(23 97 43 63 109 170 153)(24 110 98 171 44 154 64)(25 45 111 155 99 65 172)(26 100 46 66 112 173 156)(27 113 101 174 47 157 67)
(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)
G:=sub<Sym(189)| (1,48,114,158,102,68,175)(2,103,49,69,115,176,159)(3,116,104,177,50,160,70)(4,51,117,161,105,71,178)(5,106,52,72,118,179,162)(6,119,107,180,53,136,73)(7,54,120,137,108,74,181)(8,82,28,75,121,182,138)(9,122,83,183,29,139,76)(10,30,123,140,84,77,184)(11,85,31,78,124,185,141)(12,125,86,186,32,142,79)(13,33,126,143,87,80,187)(14,88,34,81,127,188,144)(15,128,89,189,35,145,55)(16,36,129,146,90,56,163)(17,91,37,57,130,164,147)(18,131,92,165,38,148,58)(19,39,132,149,93,59,166)(20,94,40,60,133,167,150)(21,134,95,168,41,151,61)(22,42,135,152,96,62,169)(23,97,43,63,109,170,153)(24,110,98,171,44,154,64)(25,45,111,155,99,65,172)(26,100,46,66,112,173,156)(27,113,101,174,47,157,67), (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)>;
G:=Group( (1,48,114,158,102,68,175)(2,103,49,69,115,176,159)(3,116,104,177,50,160,70)(4,51,117,161,105,71,178)(5,106,52,72,118,179,162)(6,119,107,180,53,136,73)(7,54,120,137,108,74,181)(8,82,28,75,121,182,138)(9,122,83,183,29,139,76)(10,30,123,140,84,77,184)(11,85,31,78,124,185,141)(12,125,86,186,32,142,79)(13,33,126,143,87,80,187)(14,88,34,81,127,188,144)(15,128,89,189,35,145,55)(16,36,129,146,90,56,163)(17,91,37,57,130,164,147)(18,131,92,165,38,148,58)(19,39,132,149,93,59,166)(20,94,40,60,133,167,150)(21,134,95,168,41,151,61)(22,42,135,152,96,62,169)(23,97,43,63,109,170,153)(24,110,98,171,44,154,64)(25,45,111,155,99,65,172)(26,100,46,66,112,173,156)(27,113,101,174,47,157,67), (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) );
G=PermutationGroup([[(1,48,114,158,102,68,175),(2,103,49,69,115,176,159),(3,116,104,177,50,160,70),(4,51,117,161,105,71,178),(5,106,52,72,118,179,162),(6,119,107,180,53,136,73),(7,54,120,137,108,74,181),(8,82,28,75,121,182,138),(9,122,83,183,29,139,76),(10,30,123,140,84,77,184),(11,85,31,78,124,185,141),(12,125,86,186,32,142,79),(13,33,126,143,87,80,187),(14,88,34,81,127,188,144),(15,128,89,189,35,145,55),(16,36,129,146,90,56,163),(17,91,37,57,130,164,147),(18,131,92,165,38,148,58),(19,39,132,149,93,59,166),(20,94,40,60,133,167,150),(21,134,95,168,41,151,61),(22,42,135,152,96,62,169),(23,97,43,63,109,170,153),(24,110,98,171,44,154,64),(25,45,111,155,99,65,172),(26,100,46,66,112,173,156),(27,113,101,174,47,157,67)], [(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)]])
C7⋊C27 is a maximal subgroup of
C7⋊C54
45 conjugacy classes
| class | 1 | 3A | 3B | 7A | 7B | 9A | ··· | 9F | 21A | 21B | 21C | 21D | 27A | ··· | 27R | 63A | ··· | 63L |
| order | 1 | 3 | 3 | 7 | 7 | 9 | ··· | 9 | 21 | 21 | 21 | 21 | 27 | ··· | 27 | 63 | ··· | 63 |
| size | 1 | 1 | 1 | 3 | 3 | 1 | ··· | 1 | 3 | 3 | 3 | 3 | 7 | ··· | 7 | 3 | ··· | 3 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 | 3 |
| type | + | ||||||
| image | C1 | C3 | C9 | C27 | C7⋊C3 | C7⋊C9 | C7⋊C27 |
| kernel | C7⋊C27 | C63 | C21 | C7 | C9 | C3 | C1 |
| # reps | 1 | 2 | 6 | 18 | 2 | 4 | 12 |
Matrix representation of C7⋊C27 ►in GL3(𝔽379) generated by
| 378 | 1 | 0 |
| 378 | 0 | 1 |
| 351 | 1 | 28 |
| 132 | 117 | 317 |
| 124 | 306 | 44 |
| 95 | 376 | 320 |
G:=sub<GL(3,GF(379))| [378,378,351,1,0,1,0,1,28],[132,124,95,117,306,376,317,44,320] >;
C7⋊C27 in GAP, Magma, Sage, TeX
C_7\rtimes C_{27} % in TeX
G:=Group("C7:C27"); // GroupNames label
G:=SmallGroup(189,1);
// by ID
G=gap.SmallGroup(189,1);
# by ID
G:=PCGroup([4,-3,-3,-3,-7,12,29,867]);
// Polycyclic
G:=Group<a,b|a^7=b^27=1,b*a*b^-1=a^4>;
// generators/relations
Export