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G = C63order 216 = 23·33

Abelian group of type [6,6,6]

direct product, abelian, monomial

Aliases: C63, SmallGroup(216,177)

Series: Derived Chief Lower central Upper central

C1 — C63
C1C3C32C33C32×C6C3×C62 — 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 [×7], C3 [×13], C22 [×7], C6 [×91], C23, C32 [×13], C2×C6 [×91], C3×C6 [×91], C22×C6 [×13], C33, C62 [×91], C32×C6 [×7], C2×C62 [×13], C3×C62 [×7], C63
Quotients: C1, C2 [×7], C3 [×13], C22 [×7], C6 [×91], C23, C32 [×13], C2×C6 [×91], C3×C6 [×91], C22×C6 [×13], C33, C62 [×91], C32×C6 [×7], C2×C62 [×13], C3×C62 [×7], C63

Smallest permutation representation of 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)
(1 156 38 35 121 161)(2 151 39 36 122 162)(3 152 40 31 123 157)(4 153 41 32 124 158)(5 154 42 33 125 159)(6 155 37 34 126 160)(7 189 66 204 130 135)(8 190 61 199 131 136)(9 191 62 200 132 137)(10 192 63 201 127 138)(11 187 64 202 128 133)(12 188 65 203 129 134)(13 174 178 49 150 103)(14 169 179 50 145 104)(15 170 180 51 146 105)(16 171 175 52 147 106)(17 172 176 53 148 107)(18 173 177 54 149 108)(19 82 118 99 113 198)(20 83 119 100 114 193)(21 84 120 101 109 194)(22 79 115 102 110 195)(23 80 116 97 111 196)(24 81 117 98 112 197)(25 77 48 70 143 165)(26 78 43 71 144 166)(27 73 44 72 139 167)(28 74 45 67 140 168)(29 75 46 68 141 163)(30 76 47 69 142 164)(55 213 181 208 96 85)(56 214 182 209 91 86)(57 215 183 210 92 87)(58 216 184 205 93 88)(59 211 185 206 94 89)(60 212 186 207 95 90)
(1 191 59 139 148 100)(2 192 60 140 149 101)(3 187 55 141 150 102)(4 188 56 142 145 97)(5 189 57 143 146 98)(6 190 58 144 147 99)(7 87 70 51 117 159)(8 88 71 52 118 160)(9 89 72 53 119 161)(10 90 67 54 120 162)(11 85 68 49 115 157)(12 86 69 50 116 158)(13 195 40 202 181 29)(14 196 41 203 182 30)(15 197 42 204 183 25)(16 198 37 199 184 26)(17 193 38 200 185 27)(18 194 39 201 186 28)(19 34 131 205 78 171)(20 35 132 206 73 172)(21 36 127 207 74 173)(22 31 128 208 75 174)(23 32 129 209 76 169)(24 33 130 210 77 170)(43 175 82 126 136 93)(44 176 83 121 137 94)(45 177 84 122 138 95)(46 178 79 123 133 96)(47 179 80 124 134 91)(48 180 81 125 135 92)(61 216 166 106 113 155)(62 211 167 107 114 156)(63 212 168 108 109 151)(64 213 163 103 110 152)(65 214 164 104 111 153)(66 215 165 105 112 154)

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), (1,156,38,35,121,161)(2,151,39,36,122,162)(3,152,40,31,123,157)(4,153,41,32,124,158)(5,154,42,33,125,159)(6,155,37,34,126,160)(7,189,66,204,130,135)(8,190,61,199,131,136)(9,191,62,200,132,137)(10,192,63,201,127,138)(11,187,64,202,128,133)(12,188,65,203,129,134)(13,174,178,49,150,103)(14,169,179,50,145,104)(15,170,180,51,146,105)(16,171,175,52,147,106)(17,172,176,53,148,107)(18,173,177,54,149,108)(19,82,118,99,113,198)(20,83,119,100,114,193)(21,84,120,101,109,194)(22,79,115,102,110,195)(23,80,116,97,111,196)(24,81,117,98,112,197)(25,77,48,70,143,165)(26,78,43,71,144,166)(27,73,44,72,139,167)(28,74,45,67,140,168)(29,75,46,68,141,163)(30,76,47,69,142,164)(55,213,181,208,96,85)(56,214,182,209,91,86)(57,215,183,210,92,87)(58,216,184,205,93,88)(59,211,185,206,94,89)(60,212,186,207,95,90), (1,191,59,139,148,100)(2,192,60,140,149,101)(3,187,55,141,150,102)(4,188,56,142,145,97)(5,189,57,143,146,98)(6,190,58,144,147,99)(7,87,70,51,117,159)(8,88,71,52,118,160)(9,89,72,53,119,161)(10,90,67,54,120,162)(11,85,68,49,115,157)(12,86,69,50,116,158)(13,195,40,202,181,29)(14,196,41,203,182,30)(15,197,42,204,183,25)(16,198,37,199,184,26)(17,193,38,200,185,27)(18,194,39,201,186,28)(19,34,131,205,78,171)(20,35,132,206,73,172)(21,36,127,207,74,173)(22,31,128,208,75,174)(23,32,129,209,76,169)(24,33,130,210,77,170)(43,175,82,126,136,93)(44,176,83,121,137,94)(45,177,84,122,138,95)(46,178,79,123,133,96)(47,179,80,124,134,91)(48,180,81,125,135,92)(61,216,166,106,113,155)(62,211,167,107,114,156)(63,212,168,108,109,151)(64,213,163,103,110,152)(65,214,164,104,111,153)(66,215,165,105,112,154)>;

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), (1,156,38,35,121,161)(2,151,39,36,122,162)(3,152,40,31,123,157)(4,153,41,32,124,158)(5,154,42,33,125,159)(6,155,37,34,126,160)(7,189,66,204,130,135)(8,190,61,199,131,136)(9,191,62,200,132,137)(10,192,63,201,127,138)(11,187,64,202,128,133)(12,188,65,203,129,134)(13,174,178,49,150,103)(14,169,179,50,145,104)(15,170,180,51,146,105)(16,171,175,52,147,106)(17,172,176,53,148,107)(18,173,177,54,149,108)(19,82,118,99,113,198)(20,83,119,100,114,193)(21,84,120,101,109,194)(22,79,115,102,110,195)(23,80,116,97,111,196)(24,81,117,98,112,197)(25,77,48,70,143,165)(26,78,43,71,144,166)(27,73,44,72,139,167)(28,74,45,67,140,168)(29,75,46,68,141,163)(30,76,47,69,142,164)(55,213,181,208,96,85)(56,214,182,209,91,86)(57,215,183,210,92,87)(58,216,184,205,93,88)(59,211,185,206,94,89)(60,212,186,207,95,90), (1,191,59,139,148,100)(2,192,60,140,149,101)(3,187,55,141,150,102)(4,188,56,142,145,97)(5,189,57,143,146,98)(6,190,58,144,147,99)(7,87,70,51,117,159)(8,88,71,52,118,160)(9,89,72,53,119,161)(10,90,67,54,120,162)(11,85,68,49,115,157)(12,86,69,50,116,158)(13,195,40,202,181,29)(14,196,41,203,182,30)(15,197,42,204,183,25)(16,198,37,199,184,26)(17,193,38,200,185,27)(18,194,39,201,186,28)(19,34,131,205,78,171)(20,35,132,206,73,172)(21,36,127,207,74,173)(22,31,128,208,75,174)(23,32,129,209,76,169)(24,33,130,210,77,170)(43,175,82,126,136,93)(44,176,83,121,137,94)(45,177,84,122,138,95)(46,178,79,123,133,96)(47,179,80,124,134,91)(48,180,81,125,135,92)(61,216,166,106,113,155)(62,211,167,107,114,156)(63,212,168,108,109,151)(64,213,163,103,110,152)(65,214,164,104,111,153)(66,215,165,105,112,154) );

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)], [(1,156,38,35,121,161),(2,151,39,36,122,162),(3,152,40,31,123,157),(4,153,41,32,124,158),(5,154,42,33,125,159),(6,155,37,34,126,160),(7,189,66,204,130,135),(8,190,61,199,131,136),(9,191,62,200,132,137),(10,192,63,201,127,138),(11,187,64,202,128,133),(12,188,65,203,129,134),(13,174,178,49,150,103),(14,169,179,50,145,104),(15,170,180,51,146,105),(16,171,175,52,147,106),(17,172,176,53,148,107),(18,173,177,54,149,108),(19,82,118,99,113,198),(20,83,119,100,114,193),(21,84,120,101,109,194),(22,79,115,102,110,195),(23,80,116,97,111,196),(24,81,117,98,112,197),(25,77,48,70,143,165),(26,78,43,71,144,166),(27,73,44,72,139,167),(28,74,45,67,140,168),(29,75,46,68,141,163),(30,76,47,69,142,164),(55,213,181,208,96,85),(56,214,182,209,91,86),(57,215,183,210,92,87),(58,216,184,205,93,88),(59,211,185,206,94,89),(60,212,186,207,95,90)], [(1,191,59,139,148,100),(2,192,60,140,149,101),(3,187,55,141,150,102),(4,188,56,142,145,97),(5,189,57,143,146,98),(6,190,58,144,147,99),(7,87,70,51,117,159),(8,88,71,52,118,160),(9,89,72,53,119,161),(10,90,67,54,120,162),(11,85,68,49,115,157),(12,86,69,50,116,158),(13,195,40,202,181,29),(14,196,41,203,182,30),(15,197,42,204,183,25),(16,198,37,199,184,26),(17,193,38,200,185,27),(18,194,39,201,186,28),(19,34,131,205,78,171),(20,35,132,206,73,172),(21,36,127,207,74,173),(22,31,128,208,75,174),(23,32,129,209,76,169),(24,33,130,210,77,170),(43,175,82,126,136,93),(44,176,83,121,137,94),(45,177,84,122,138,95),(46,178,79,123,133,96),(47,179,80,124,134,91),(48,180,81,125,135,92),(61,216,166,106,113,155),(62,211,167,107,114,156),(63,212,168,108,109,151),(64,213,163,103,110,152),(65,214,164,104,111,153),(66,215,165,105,112,154)])

C63 is a maximal subgroup of   C63.C2

216 conjugacy classes

class 1 2A···2G3A···3Z6A···6FZ
order12···23···36···6
size11···11···11···1

216 irreducible representations

dim1111
type++
imageC1C2C3C6
kernelC63C3×C62C2×C62C62
# reps1726182

Matrix representation of C63 in GL3(𝔽7) generated by

500
040
002
,
600
060
002
,
600
020
003
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

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