direct product, abelian, monomial, 3-elementary
Aliases: C3×C63, SmallGroup(189,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C63 |
| C1 — C3×C63 |
| C1 — C3×C63 |
Generators and relations for C3×C63
G = < a,b | a3=b63=1, ab=ba >
Smallest permutation representation of C3×C63
►Regular action on 189 points
Generators in S189
(1 84 189)(2 85 127)(3 86 128)(4 87 129)(5 88 130)(6 89 131)(7 90 132)(8 91 133)(9 92 134)(10 93 135)(11 94 136)(12 95 137)(13 96 138)(14 97 139)(15 98 140)(16 99 141)(17 100 142)(18 101 143)(19 102 144)(20 103 145)(21 104 146)(22 105 147)(23 106 148)(24 107 149)(25 108 150)(26 109 151)(27 110 152)(28 111 153)(29 112 154)(30 113 155)(31 114 156)(32 115 157)(33 116 158)(34 117 159)(35 118 160)(36 119 161)(37 120 162)(38 121 163)(39 122 164)(40 123 165)(41 124 166)(42 125 167)(43 126 168)(44 64 169)(45 65 170)(46 66 171)(47 67 172)(48 68 173)(49 69 174)(50 70 175)(51 71 176)(52 72 177)(53 73 178)(54 74 179)(55 75 180)(56 76 181)(57 77 182)(58 78 183)(59 79 184)(60 80 185)(61 81 186)(62 82 187)(63 83 188)
(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)| (1,84,189)(2,85,127)(3,86,128)(4,87,129)(5,88,130)(6,89,131)(7,90,132)(8,91,133)(9,92,134)(10,93,135)(11,94,136)(12,95,137)(13,96,138)(14,97,139)(15,98,140)(16,99,141)(17,100,142)(18,101,143)(19,102,144)(20,103,145)(21,104,146)(22,105,147)(23,106,148)(24,107,149)(25,108,150)(26,109,151)(27,110,152)(28,111,153)(29,112,154)(30,113,155)(31,114,156)(32,115,157)(33,116,158)(34,117,159)(35,118,160)(36,119,161)(37,120,162)(38,121,163)(39,122,164)(40,123,165)(41,124,166)(42,125,167)(43,126,168)(44,64,169)(45,65,170)(46,66,171)(47,67,172)(48,68,173)(49,69,174)(50,70,175)(51,71,176)(52,72,177)(53,73,178)(54,74,179)(55,75,180)(56,76,181)(57,77,182)(58,78,183)(59,79,184)(60,80,185)(61,81,186)(62,82,187)(63,83,188), (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( (1,84,189)(2,85,127)(3,86,128)(4,87,129)(5,88,130)(6,89,131)(7,90,132)(8,91,133)(9,92,134)(10,93,135)(11,94,136)(12,95,137)(13,96,138)(14,97,139)(15,98,140)(16,99,141)(17,100,142)(18,101,143)(19,102,144)(20,103,145)(21,104,146)(22,105,147)(23,106,148)(24,107,149)(25,108,150)(26,109,151)(27,110,152)(28,111,153)(29,112,154)(30,113,155)(31,114,156)(32,115,157)(33,116,158)(34,117,159)(35,118,160)(36,119,161)(37,120,162)(38,121,163)(39,122,164)(40,123,165)(41,124,166)(42,125,167)(43,126,168)(44,64,169)(45,65,170)(46,66,171)(47,67,172)(48,68,173)(49,69,174)(50,70,175)(51,71,176)(52,72,177)(53,73,178)(54,74,179)(55,75,180)(56,76,181)(57,77,182)(58,78,183)(59,79,184)(60,80,185)(61,81,186)(62,82,187)(63,83,188), (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([[(1,84,189),(2,85,127),(3,86,128),(4,87,129),(5,88,130),(6,89,131),(7,90,132),(8,91,133),(9,92,134),(10,93,135),(11,94,136),(12,95,137),(13,96,138),(14,97,139),(15,98,140),(16,99,141),(17,100,142),(18,101,143),(19,102,144),(20,103,145),(21,104,146),(22,105,147),(23,106,148),(24,107,149),(25,108,150),(26,109,151),(27,110,152),(28,111,153),(29,112,154),(30,113,155),(31,114,156),(32,115,157),(33,116,158),(34,117,159),(35,118,160),(36,119,161),(37,120,162),(38,121,163),(39,122,164),(40,123,165),(41,124,166),(42,125,167),(43,126,168),(44,64,169),(45,65,170),(46,66,171),(47,67,172),(48,68,173),(49,69,174),(50,70,175),(51,71,176),(52,72,177),(53,73,178),(54,74,179),(55,75,180),(56,76,181),(57,77,182),(58,78,183),(59,79,184),(60,80,185),(61,81,186),(62,82,187),(63,83,188)], [(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)]])
C3×C63 is a maximal subgroup of
C3⋊D63
189 conjugacy classes
| class | 1 | 3A | ··· | 3H | 7A | ··· | 7F | 9A | ··· | 9R | 21A | ··· | 21AV | 63A | ··· | 63DD |
| order | 1 | 3 | ··· | 3 | 7 | ··· | 7 | 9 | ··· | 9 | 21 | ··· | 21 | 63 | ··· | 63 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
189 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | |||||||
| image | C1 | C3 | C3 | C7 | C9 | C21 | C21 | C63 |
| kernel | C3×C63 | C63 | C3×C21 | C3×C9 | C21 | C9 | C32 | C3 |
| # reps | 1 | 6 | 2 | 6 | 18 | 36 | 12 | 108 |
Matrix representation of C3×C63 ►in GL2(𝔽127) generated by
| 107 | 0 |
| 0 | 107 |
| 107 | 0 |
| 0 | 44 |
G:=sub<GL(2,GF(127))| [107,0,0,107],[107,0,0,44] >;
C3×C63 in GAP, Magma, Sage, TeX
C_3\times C_{63} % in TeX
G:=Group("C3xC63"); // GroupNames label
G:=SmallGroup(189,9);
// by ID
G=gap.SmallGroup(189,9);
# by ID
G:=PCGroup([4,-3,-3,-7,-3,252]);
// Polycyclic
G:=Group<a,b|a^3=b^63=1,a*b=b*a>;
// generators/relations
Export