direct product, abelian, monomial
Aliases: C63, SmallGroup(216,177)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C63 |
| C1 — C63 |
| C1 — C63 |
Generators and relations for C63
G = < a,b,c | a6=b6=c6=1, ab=ba, ac=ca, bc=cb >
Subgroups: 448, all normal (4 characteristic)
C1, C2, C3, C22, C6, C23, C32, C2×C6, C3×C6, C22×C6, C33, C62, C32×C6, C2×C62, C3×C62, C63
Quotients: C1, C2, C3, C22, C6, C23, C32, C2×C6, C3×C6, C22×C6, C33, C62, C32×C6, C2×C62, C3×C62, C63
►Regular action on 216 points
Generators in S216
(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)
(1 75 167 68 27 206)(2 76 168 69 28 207)(3 77 163 70 29 208)(4 78 164 71 30 209)(5 73 165 72 25 210)(6 74 166 67 26 205)(7 79 115 105 20 102)(8 80 116 106 21 97)(9 81 117 107 22 98)(10 82 118 108 23 99)(11 83 119 103 24 100)(12 84 120 104 19 101)(13 121 42 139 51 154)(14 122 37 140 52 155)(15 123 38 141 53 156)(16 124 39 142 54 151)(17 125 40 143 49 152)(18 126 41 144 50 153)(31 132 204 180 191 112)(32 127 199 175 192 113)(33 128 200 176 187 114)(34 129 201 177 188 109)(35 130 202 178 189 110)(36 131 203 179 190 111)(43 158 133 62 149 172)(44 159 134 63 150 173)(45 160 135 64 145 174)(46 161 136 65 146 169)(47 162 137 66 147 170)(48 157 138 61 148 171)(55 213 181 193 96 85)(56 214 182 194 91 86)(57 215 183 195 92 87)(58 216 184 196 93 88)(59 211 185 197 94 89)(60 212 186 198 95 90)
(1 133 59 187 17 103)(2 134 60 188 18 104)(3 135 55 189 13 105)(4 136 56 190 14 106)(5 137 57 191 15 107)(6 138 58 192 16 108)(7 70 174 193 130 139)(8 71 169 194 131 140)(9 72 170 195 132 141)(10 67 171 196 127 142)(11 68 172 197 128 143)(12 69 173 198 129 144)(19 76 63 212 109 126)(20 77 64 213 110 121)(21 78 65 214 111 122)(22 73 66 215 112 123)(23 74 61 216 113 124)(24 75 62 211 114 125)(25 47 92 204 53 81)(26 48 93 199 54 82)(27 43 94 200 49 83)(28 44 95 201 50 84)(29 45 96 202 51 79)(30 46 91 203 52 80)(31 38 98 165 147 183)(32 39 99 166 148 184)(33 40 100 167 149 185)(34 41 101 168 150 186)(35 42 102 163 145 181)(36 37 97 164 146 182)(85 178 154 115 208 160)(86 179 155 116 209 161)(87 180 156 117 210 162)(88 175 151 118 205 157)(89 176 152 119 206 158)(90 177 153 120 207 159)
G:=sub<Sym(216)| (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), (1,75,167,68,27,206)(2,76,168,69,28,207)(3,77,163,70,29,208)(4,78,164,71,30,209)(5,73,165,72,25,210)(6,74,166,67,26,205)(7,79,115,105,20,102)(8,80,116,106,21,97)(9,81,117,107,22,98)(10,82,118,108,23,99)(11,83,119,103,24,100)(12,84,120,104,19,101)(13,121,42,139,51,154)(14,122,37,140,52,155)(15,123,38,141,53,156)(16,124,39,142,54,151)(17,125,40,143,49,152)(18,126,41,144,50,153)(31,132,204,180,191,112)(32,127,199,175,192,113)(33,128,200,176,187,114)(34,129,201,177,188,109)(35,130,202,178,189,110)(36,131,203,179,190,111)(43,158,133,62,149,172)(44,159,134,63,150,173)(45,160,135,64,145,174)(46,161,136,65,146,169)(47,162,137,66,147,170)(48,157,138,61,148,171)(55,213,181,193,96,85)(56,214,182,194,91,86)(57,215,183,195,92,87)(58,216,184,196,93,88)(59,211,185,197,94,89)(60,212,186,198,95,90), (1,133,59,187,17,103)(2,134,60,188,18,104)(3,135,55,189,13,105)(4,136,56,190,14,106)(5,137,57,191,15,107)(6,138,58,192,16,108)(7,70,174,193,130,139)(8,71,169,194,131,140)(9,72,170,195,132,141)(10,67,171,196,127,142)(11,68,172,197,128,143)(12,69,173,198,129,144)(19,76,63,212,109,126)(20,77,64,213,110,121)(21,78,65,214,111,122)(22,73,66,215,112,123)(23,74,61,216,113,124)(24,75,62,211,114,125)(25,47,92,204,53,81)(26,48,93,199,54,82)(27,43,94,200,49,83)(28,44,95,201,50,84)(29,45,96,202,51,79)(30,46,91,203,52,80)(31,38,98,165,147,183)(32,39,99,166,148,184)(33,40,100,167,149,185)(34,41,101,168,150,186)(35,42,102,163,145,181)(36,37,97,164,146,182)(85,178,154,115,208,160)(86,179,155,116,209,161)(87,180,156,117,210,162)(88,175,151,118,205,157)(89,176,152,119,206,158)(90,177,153,120,207,159)>;
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), (1,75,167,68,27,206)(2,76,168,69,28,207)(3,77,163,70,29,208)(4,78,164,71,30,209)(5,73,165,72,25,210)(6,74,166,67,26,205)(7,79,115,105,20,102)(8,80,116,106,21,97)(9,81,117,107,22,98)(10,82,118,108,23,99)(11,83,119,103,24,100)(12,84,120,104,19,101)(13,121,42,139,51,154)(14,122,37,140,52,155)(15,123,38,141,53,156)(16,124,39,142,54,151)(17,125,40,143,49,152)(18,126,41,144,50,153)(31,132,204,180,191,112)(32,127,199,175,192,113)(33,128,200,176,187,114)(34,129,201,177,188,109)(35,130,202,178,189,110)(36,131,203,179,190,111)(43,158,133,62,149,172)(44,159,134,63,150,173)(45,160,135,64,145,174)(46,161,136,65,146,169)(47,162,137,66,147,170)(48,157,138,61,148,171)(55,213,181,193,96,85)(56,214,182,194,91,86)(57,215,183,195,92,87)(58,216,184,196,93,88)(59,211,185,197,94,89)(60,212,186,198,95,90), (1,133,59,187,17,103)(2,134,60,188,18,104)(3,135,55,189,13,105)(4,136,56,190,14,106)(5,137,57,191,15,107)(6,138,58,192,16,108)(7,70,174,193,130,139)(8,71,169,194,131,140)(9,72,170,195,132,141)(10,67,171,196,127,142)(11,68,172,197,128,143)(12,69,173,198,129,144)(19,76,63,212,109,126)(20,77,64,213,110,121)(21,78,65,214,111,122)(22,73,66,215,112,123)(23,74,61,216,113,124)(24,75,62,211,114,125)(25,47,92,204,53,81)(26,48,93,199,54,82)(27,43,94,200,49,83)(28,44,95,201,50,84)(29,45,96,202,51,79)(30,46,91,203,52,80)(31,38,98,165,147,183)(32,39,99,166,148,184)(33,40,100,167,149,185)(34,41,101,168,150,186)(35,42,102,163,145,181)(36,37,97,164,146,182)(85,178,154,115,208,160)(86,179,155,116,209,161)(87,180,156,117,210,162)(88,175,151,118,205,157)(89,176,152,119,206,158)(90,177,153,120,207,159) );
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)], [(1,75,167,68,27,206),(2,76,168,69,28,207),(3,77,163,70,29,208),(4,78,164,71,30,209),(5,73,165,72,25,210),(6,74,166,67,26,205),(7,79,115,105,20,102),(8,80,116,106,21,97),(9,81,117,107,22,98),(10,82,118,108,23,99),(11,83,119,103,24,100),(12,84,120,104,19,101),(13,121,42,139,51,154),(14,122,37,140,52,155),(15,123,38,141,53,156),(16,124,39,142,54,151),(17,125,40,143,49,152),(18,126,41,144,50,153),(31,132,204,180,191,112),(32,127,199,175,192,113),(33,128,200,176,187,114),(34,129,201,177,188,109),(35,130,202,178,189,110),(36,131,203,179,190,111),(43,158,133,62,149,172),(44,159,134,63,150,173),(45,160,135,64,145,174),(46,161,136,65,146,169),(47,162,137,66,147,170),(48,157,138,61,148,171),(55,213,181,193,96,85),(56,214,182,194,91,86),(57,215,183,195,92,87),(58,216,184,196,93,88),(59,211,185,197,94,89),(60,212,186,198,95,90)], [(1,133,59,187,17,103),(2,134,60,188,18,104),(3,135,55,189,13,105),(4,136,56,190,14,106),(5,137,57,191,15,107),(6,138,58,192,16,108),(7,70,174,193,130,139),(8,71,169,194,131,140),(9,72,170,195,132,141),(10,67,171,196,127,142),(11,68,172,197,128,143),(12,69,173,198,129,144),(19,76,63,212,109,126),(20,77,64,213,110,121),(21,78,65,214,111,122),(22,73,66,215,112,123),(23,74,61,216,113,124),(24,75,62,211,114,125),(25,47,92,204,53,81),(26,48,93,199,54,82),(27,43,94,200,49,83),(28,44,95,201,50,84),(29,45,96,202,51,79),(30,46,91,203,52,80),(31,38,98,165,147,183),(32,39,99,166,148,184),(33,40,100,167,149,185),(34,41,101,168,150,186),(35,42,102,163,145,181),(36,37,97,164,146,182),(85,178,154,115,208,160),(86,179,155,116,209,161),(87,180,156,117,210,162),(88,175,151,118,205,157),(89,176,152,119,206,158),(90,177,153,120,207,159)]])
C63 is a maximal subgroup of
C63.C2
216 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | ··· | 3Z | 6A | ··· | 6FZ |
| order | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 6 | ··· | 6 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
216 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | + | ||
| image | C1 | C2 | C3 | C6 |
| kernel | C63 | C3×C62 | C2×C62 | C62 |
| # reps | 1 | 7 | 26 | 182 |
Matrix representation of C63 ►in GL3(𝔽7) generated by
| 5 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 2 |
| 6 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 0 | 2 |
| 6 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 3 |
G:=sub<GL(3,GF(7))| [5,0,0,0,4,0,0,0,2],[6,0,0,0,6,0,0,0,2],[6,0,0,0,2,0,0,0,3] >;
C63 in GAP, Magma, Sage, TeX
C_6^3
% in TeX
G:=Group("C6^3"); // GroupNames label
G:=SmallGroup(216,177);
// by ID
G=gap.SmallGroup(216,177);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-3]);
// Polycyclic
G:=Group<a,b,c|a^6=b^6=c^6=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations