metacyclic, supersoluble, monomial, Z-group
Aliases: C7⋊C54, D7⋊C27, C9.F7, C63.C6, C21.C18, C7⋊C27⋊C2, C3.(C7⋊C18), (C9×D7).C3, (C3×D7).C9, SmallGroup(378,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C7⋊C54 |
Generators and relations for C7⋊C54
G = < a,b | a7=b54=1, bab-1=a3 >
Smallest permutation representation of C7⋊C54
►On 189 points
Generators in S189
(1 53 150 96 123 177 80)(2 97 81 151 178 54 124)(3 152 125 28 55 98 179)(4 29 180 126 99 153 56)(5 127 57 181 154 30 100)(6 182 101 58 31 128 155)(7 59 156 102 129 183 32)(8 103 33 157 184 60 130)(9 158 131 34 61 104 185)(10 35 186 132 105 159 62)(11 133 63 187 160 36 106)(12 188 107 64 37 134 161)(13 65 162 108 135 189 38)(14 109 39 163 136 66 82)(15 164 83 40 67 110 137)(16 41 138 84 111 165 68)(17 85 69 139 166 42 112)(18 140 113 70 43 86 167)(19 71 168 114 87 141 44)(20 115 45 169 142 72 88)(21 170 89 46 73 116 143)(22 47 144 90 117 171 74)(23 91 75 145 172 48 118)(24 146 119 76 49 92 173)(25 77 174 120 93 147 50)(26 121 51 175 148 78 94)(27 176 95 52 79 122 149)
(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)| (1,53,150,96,123,177,80)(2,97,81,151,178,54,124)(3,152,125,28,55,98,179)(4,29,180,126,99,153,56)(5,127,57,181,154,30,100)(6,182,101,58,31,128,155)(7,59,156,102,129,183,32)(8,103,33,157,184,60,130)(9,158,131,34,61,104,185)(10,35,186,132,105,159,62)(11,133,63,187,160,36,106)(12,188,107,64,37,134,161)(13,65,162,108,135,189,38)(14,109,39,163,136,66,82)(15,164,83,40,67,110,137)(16,41,138,84,111,165,68)(17,85,69,139,166,42,112)(18,140,113,70,43,86,167)(19,71,168,114,87,141,44)(20,115,45,169,142,72,88)(21,170,89,46,73,116,143)(22,47,144,90,117,171,74)(23,91,75,145,172,48,118)(24,146,119,76,49,92,173)(25,77,174,120,93,147,50)(26,121,51,175,148,78,94)(27,176,95,52,79,122,149), (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( (1,53,150,96,123,177,80)(2,97,81,151,178,54,124)(3,152,125,28,55,98,179)(4,29,180,126,99,153,56)(5,127,57,181,154,30,100)(6,182,101,58,31,128,155)(7,59,156,102,129,183,32)(8,103,33,157,184,60,130)(9,158,131,34,61,104,185)(10,35,186,132,105,159,62)(11,133,63,187,160,36,106)(12,188,107,64,37,134,161)(13,65,162,108,135,189,38)(14,109,39,163,136,66,82)(15,164,83,40,67,110,137)(16,41,138,84,111,165,68)(17,85,69,139,166,42,112)(18,140,113,70,43,86,167)(19,71,168,114,87,141,44)(20,115,45,169,142,72,88)(21,170,89,46,73,116,143)(22,47,144,90,117,171,74)(23,91,75,145,172,48,118)(24,146,119,76,49,92,173)(25,77,174,120,93,147,50)(26,121,51,175,148,78,94)(27,176,95,52,79,122,149), (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([[(1,53,150,96,123,177,80),(2,97,81,151,178,54,124),(3,152,125,28,55,98,179),(4,29,180,126,99,153,56),(5,127,57,181,154,30,100),(6,182,101,58,31,128,155),(7,59,156,102,129,183,32),(8,103,33,157,184,60,130),(9,158,131,34,61,104,185),(10,35,186,132,105,159,62),(11,133,63,187,160,36,106),(12,188,107,64,37,134,161),(13,65,162,108,135,189,38),(14,109,39,163,136,66,82),(15,164,83,40,67,110,137),(16,41,138,84,111,165,68),(17,85,69,139,166,42,112),(18,140,113,70,43,86,167),(19,71,168,114,87,141,44),(20,115,45,169,142,72,88),(21,170,89,46,73,116,143),(22,47,144,90,117,171,74),(23,91,75,145,172,48,118),(24,146,119,76,49,92,173),(25,77,174,120,93,147,50),(26,121,51,175,148,78,94),(27,176,95,52,79,122,149)], [(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)]])
63 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 7 | 9A | ··· | 9F | 18A | ··· | 18F | 21A | 21B | 27A | ··· | 27R | 54A | ··· | 54R | 63A | ··· | 63F |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 7 | 9 | ··· | 9 | 18 | ··· | 18 | 21 | 21 | 27 | ··· | 27 | 54 | ··· | 54 | 63 | ··· | 63 |
| size | 1 | 7 | 1 | 1 | 7 | 7 | 6 | 1 | ··· | 1 | 7 | ··· | 7 | 6 | 6 | 7 | ··· | 7 | 7 | ··· | 7 | 6 | ··· | 6 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 6 | 6 |
| type | + | + | + | ||||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | C27 | C54 | F7 | C7⋊C18 | C7⋊C54 |
| kernel | C7⋊C54 | C7⋊C27 | C9×D7 | C63 | C3×D7 | C21 | D7 | C7 | C9 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 18 | 18 | 1 | 2 | 6 |
Matrix representation of C7⋊C54 ►in GL7(𝔽379)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 378 | 1 | 0 | 0 | 0 | 0 |
| 0 | 378 | 0 | 1 | 0 | 0 | 0 |
| 0 | 378 | 0 | 0 | 1 | 0 | 0 |
| 0 | 378 | 0 | 0 | 0 | 1 | 0 |
| 0 | 378 | 0 | 0 | 0 | 0 | 1 |
| 0 | 378 | 0 | 0 | 0 | 0 | 0 |
| 228 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 103 | 276 | 183 | 208 | 0 | 171 |
| 0 | 103 | 68 | 0 | 311 | 276 | 354 |
| 0 | 0 | 251 | 208 | 311 | 68 | 171 |
| 0 | 171 | 68 | 311 | 208 | 251 | 0 |
| 0 | 354 | 276 | 311 | 0 | 68 | 103 |
| 0 | 171 | 0 | 208 | 183 | 276 | 103 |
G:=sub<GL(7,GF(379))| [1,0,0,0,0,0,0,0,378,378,378,378,378,378,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0],[228,0,0,0,0,0,0,0,103,103,0,171,354,171,0,276,68,251,68,276,0,0,183,0,208,311,311,208,0,208,311,311,208,0,183,0,0,276,68,251,68,276,0,171,354,171,0,103,103] >;
C7⋊C54 in GAP, Magma, Sage, TeX
C_7\rtimes C_{54} % in TeX
G:=Group("C7:C54"); // GroupNames label
G:=SmallGroup(378,1);
// by ID
G=gap.SmallGroup(378,1);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-7,36,57,8104,2709]);
// Polycyclic
G:=Group<a,b|a^7=b^54=1,b*a*b^-1=a^3>;
// generators/relations
Export