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## G = C3×C92order 243 = 35

### Abelian group of type [3,9,9]

Aliases: C3×C92, SmallGroup(243,31)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C3×C92
 Chief series C1 — C3 — C32 — C33 — C32×C9 — C3×C92
 Lower central C1 — C3×C92
 Upper central C1 — C3×C92
 Jennings C1 — C32 — C32 — C3×C92

Generators and relations for C3×C92
G = < a,b,c | a3=b9=c9=1, ab=ba, ac=ca, bc=cb >

Subgroups: 126, all normal (4 characteristic)
C1, C3 [×13], C9 [×36], C32, C32 [×12], C3×C9 [×48], C33, C92 [×9], C32×C9 [×4], C3×C92
Quotients: C1, C3 [×13], C9 [×36], C32 [×13], C3×C9 [×48], C33, C92 [×9], C32×C9 [×4], C3×C92

Smallest permutation representation of C3×C92
Regular action on 243 points
Generators in S243
(1 48 107)(2 49 108)(3 50 100)(4 51 101)(5 52 102)(6 53 103)(7 54 104)(8 46 105)(9 47 106)(10 152 227)(11 153 228)(12 145 229)(13 146 230)(14 147 231)(15 148 232)(16 149 233)(17 150 234)(18 151 226)(19 161 200)(20 162 201)(21 154 202)(22 155 203)(23 156 204)(24 157 205)(25 158 206)(26 159 207)(27 160 199)(28 87 209)(29 88 210)(30 89 211)(31 90 212)(32 82 213)(33 83 214)(34 84 215)(35 85 216)(36 86 208)(37 96 218)(38 97 219)(39 98 220)(40 99 221)(41 91 222)(42 92 223)(43 93 224)(44 94 225)(45 95 217)(55 117 183)(56 109 184)(57 110 185)(58 111 186)(59 112 187)(60 113 188)(61 114 189)(62 115 181)(63 116 182)(64 124 163)(65 125 164)(66 126 165)(67 118 166)(68 119 167)(69 120 168)(70 121 169)(71 122 170)(72 123 171)(73 135 174)(74 127 175)(75 128 176)(76 129 177)(77 130 178)(78 131 179)(79 132 180)(80 133 172)(81 134 173)(136 195 236)(137 196 237)(138 197 238)(139 198 239)(140 190 240)(141 191 241)(142 192 242)(143 193 243)(144 194 235)
(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)(217 218 219 220 221 222 223 224 225)(226 227 228 229 230 231 232 233 234)(235 236 237 238 239 240 241 242 243)
(1 65 135 215 138 98 116 232 158)(2 66 127 216 139 99 117 233 159)(3 67 128 208 140 91 109 234 160)(4 68 129 209 141 92 110 226 161)(5 69 130 210 142 93 111 227 162)(6 70 131 211 143 94 112 228 154)(7 71 132 212 144 95 113 229 155)(8 72 133 213 136 96 114 230 156)(9 64 134 214 137 97 115 231 157)(10 201 52 120 178 29 192 224 186)(11 202 53 121 179 30 193 225 187)(12 203 54 122 180 31 194 217 188)(13 204 46 123 172 32 195 218 189)(14 205 47 124 173 33 196 219 181)(15 206 48 125 174 34 197 220 182)(16 207 49 126 175 35 198 221 183)(17 199 50 118 176 36 190 222 184)(18 200 51 119 177 28 191 223 185)(19 101 167 76 87 241 42 57 151)(20 102 168 77 88 242 43 58 152)(21 103 169 78 89 243 44 59 153)(22 104 170 79 90 235 45 60 145)(23 105 171 80 82 236 37 61 146)(24 106 163 81 83 237 38 62 147)(25 107 164 73 84 238 39 63 148)(26 108 165 74 85 239 40 55 149)(27 100 166 75 86 240 41 56 150)

G:=sub<Sym(243)| (1,48,107)(2,49,108)(3,50,100)(4,51,101)(5,52,102)(6,53,103)(7,54,104)(8,46,105)(9,47,106)(10,152,227)(11,153,228)(12,145,229)(13,146,230)(14,147,231)(15,148,232)(16,149,233)(17,150,234)(18,151,226)(19,161,200)(20,162,201)(21,154,202)(22,155,203)(23,156,204)(24,157,205)(25,158,206)(26,159,207)(27,160,199)(28,87,209)(29,88,210)(30,89,211)(31,90,212)(32,82,213)(33,83,214)(34,84,215)(35,85,216)(36,86,208)(37,96,218)(38,97,219)(39,98,220)(40,99,221)(41,91,222)(42,92,223)(43,93,224)(44,94,225)(45,95,217)(55,117,183)(56,109,184)(57,110,185)(58,111,186)(59,112,187)(60,113,188)(61,114,189)(62,115,181)(63,116,182)(64,124,163)(65,125,164)(66,126,165)(67,118,166)(68,119,167)(69,120,168)(70,121,169)(71,122,170)(72,123,171)(73,135,174)(74,127,175)(75,128,176)(76,129,177)(77,130,178)(78,131,179)(79,132,180)(80,133,172)(81,134,173)(136,195,236)(137,196,237)(138,197,238)(139,198,239)(140,190,240)(141,191,241)(142,192,242)(143,193,243)(144,194,235), (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)(217,218,219,220,221,222,223,224,225)(226,227,228,229,230,231,232,233,234)(235,236,237,238,239,240,241,242,243), (1,65,135,215,138,98,116,232,158)(2,66,127,216,139,99,117,233,159)(3,67,128,208,140,91,109,234,160)(4,68,129,209,141,92,110,226,161)(5,69,130,210,142,93,111,227,162)(6,70,131,211,143,94,112,228,154)(7,71,132,212,144,95,113,229,155)(8,72,133,213,136,96,114,230,156)(9,64,134,214,137,97,115,231,157)(10,201,52,120,178,29,192,224,186)(11,202,53,121,179,30,193,225,187)(12,203,54,122,180,31,194,217,188)(13,204,46,123,172,32,195,218,189)(14,205,47,124,173,33,196,219,181)(15,206,48,125,174,34,197,220,182)(16,207,49,126,175,35,198,221,183)(17,199,50,118,176,36,190,222,184)(18,200,51,119,177,28,191,223,185)(19,101,167,76,87,241,42,57,151)(20,102,168,77,88,242,43,58,152)(21,103,169,78,89,243,44,59,153)(22,104,170,79,90,235,45,60,145)(23,105,171,80,82,236,37,61,146)(24,106,163,81,83,237,38,62,147)(25,107,164,73,84,238,39,63,148)(26,108,165,74,85,239,40,55,149)(27,100,166,75,86,240,41,56,150)>;

G:=Group( (1,48,107)(2,49,108)(3,50,100)(4,51,101)(5,52,102)(6,53,103)(7,54,104)(8,46,105)(9,47,106)(10,152,227)(11,153,228)(12,145,229)(13,146,230)(14,147,231)(15,148,232)(16,149,233)(17,150,234)(18,151,226)(19,161,200)(20,162,201)(21,154,202)(22,155,203)(23,156,204)(24,157,205)(25,158,206)(26,159,207)(27,160,199)(28,87,209)(29,88,210)(30,89,211)(31,90,212)(32,82,213)(33,83,214)(34,84,215)(35,85,216)(36,86,208)(37,96,218)(38,97,219)(39,98,220)(40,99,221)(41,91,222)(42,92,223)(43,93,224)(44,94,225)(45,95,217)(55,117,183)(56,109,184)(57,110,185)(58,111,186)(59,112,187)(60,113,188)(61,114,189)(62,115,181)(63,116,182)(64,124,163)(65,125,164)(66,126,165)(67,118,166)(68,119,167)(69,120,168)(70,121,169)(71,122,170)(72,123,171)(73,135,174)(74,127,175)(75,128,176)(76,129,177)(77,130,178)(78,131,179)(79,132,180)(80,133,172)(81,134,173)(136,195,236)(137,196,237)(138,197,238)(139,198,239)(140,190,240)(141,191,241)(142,192,242)(143,193,243)(144,194,235), (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)(217,218,219,220,221,222,223,224,225)(226,227,228,229,230,231,232,233,234)(235,236,237,238,239,240,241,242,243), (1,65,135,215,138,98,116,232,158)(2,66,127,216,139,99,117,233,159)(3,67,128,208,140,91,109,234,160)(4,68,129,209,141,92,110,226,161)(5,69,130,210,142,93,111,227,162)(6,70,131,211,143,94,112,228,154)(7,71,132,212,144,95,113,229,155)(8,72,133,213,136,96,114,230,156)(9,64,134,214,137,97,115,231,157)(10,201,52,120,178,29,192,224,186)(11,202,53,121,179,30,193,225,187)(12,203,54,122,180,31,194,217,188)(13,204,46,123,172,32,195,218,189)(14,205,47,124,173,33,196,219,181)(15,206,48,125,174,34,197,220,182)(16,207,49,126,175,35,198,221,183)(17,199,50,118,176,36,190,222,184)(18,200,51,119,177,28,191,223,185)(19,101,167,76,87,241,42,57,151)(20,102,168,77,88,242,43,58,152)(21,103,169,78,89,243,44,59,153)(22,104,170,79,90,235,45,60,145)(23,105,171,80,82,236,37,61,146)(24,106,163,81,83,237,38,62,147)(25,107,164,73,84,238,39,63,148)(26,108,165,74,85,239,40,55,149)(27,100,166,75,86,240,41,56,150) );

G=PermutationGroup([(1,48,107),(2,49,108),(3,50,100),(4,51,101),(5,52,102),(6,53,103),(7,54,104),(8,46,105),(9,47,106),(10,152,227),(11,153,228),(12,145,229),(13,146,230),(14,147,231),(15,148,232),(16,149,233),(17,150,234),(18,151,226),(19,161,200),(20,162,201),(21,154,202),(22,155,203),(23,156,204),(24,157,205),(25,158,206),(26,159,207),(27,160,199),(28,87,209),(29,88,210),(30,89,211),(31,90,212),(32,82,213),(33,83,214),(34,84,215),(35,85,216),(36,86,208),(37,96,218),(38,97,219),(39,98,220),(40,99,221),(41,91,222),(42,92,223),(43,93,224),(44,94,225),(45,95,217),(55,117,183),(56,109,184),(57,110,185),(58,111,186),(59,112,187),(60,113,188),(61,114,189),(62,115,181),(63,116,182),(64,124,163),(65,125,164),(66,126,165),(67,118,166),(68,119,167),(69,120,168),(70,121,169),(71,122,170),(72,123,171),(73,135,174),(74,127,175),(75,128,176),(76,129,177),(77,130,178),(78,131,179),(79,132,180),(80,133,172),(81,134,173),(136,195,236),(137,196,237),(138,197,238),(139,198,239),(140,190,240),(141,191,241),(142,192,242),(143,193,243),(144,194,235)], [(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),(217,218,219,220,221,222,223,224,225),(226,227,228,229,230,231,232,233,234),(235,236,237,238,239,240,241,242,243)], [(1,65,135,215,138,98,116,232,158),(2,66,127,216,139,99,117,233,159),(3,67,128,208,140,91,109,234,160),(4,68,129,209,141,92,110,226,161),(5,69,130,210,142,93,111,227,162),(6,70,131,211,143,94,112,228,154),(7,71,132,212,144,95,113,229,155),(8,72,133,213,136,96,114,230,156),(9,64,134,214,137,97,115,231,157),(10,201,52,120,178,29,192,224,186),(11,202,53,121,179,30,193,225,187),(12,203,54,122,180,31,194,217,188),(13,204,46,123,172,32,195,218,189),(14,205,47,124,173,33,196,219,181),(15,206,48,125,174,34,197,220,182),(16,207,49,126,175,35,198,221,183),(17,199,50,118,176,36,190,222,184),(18,200,51,119,177,28,191,223,185),(19,101,167,76,87,241,42,57,151),(20,102,168,77,88,242,43,58,152),(21,103,169,78,89,243,44,59,153),(22,104,170,79,90,235,45,60,145),(23,105,171,80,82,236,37,61,146),(24,106,163,81,83,237,38,62,147),(25,107,164,73,84,238,39,63,148),(26,108,165,74,85,239,40,55,149),(27,100,166,75,86,240,41,56,150)])

C3×C92 is a maximal subgroup of   C928S3

243 conjugacy classes

 class 1 3A ··· 3Z 9A ··· 9HH order 1 3 ··· 3 9 ··· 9 size 1 1 ··· 1 1 ··· 1

243 irreducible representations

 dim 1 1 1 1 type + image C1 C3 C3 C9 kernel C3×C92 C92 C32×C9 C3×C9 # reps 1 18 8 216

Matrix representation of C3×C92 in GL3(𝔽19) generated by

 11 0 0 0 11 0 0 0 7
,
 1 0 0 0 16 0 0 0 1
,
 1 0 0 0 5 0 0 0 17
G:=sub<GL(3,GF(19))| [11,0,0,0,11,0,0,0,7],[1,0,0,0,16,0,0,0,1],[1,0,0,0,5,0,0,0,17] >;

C3×C92 in GAP, Magma, Sage, TeX

C_3\times C_9^2
% in TeX

G:=Group("C3xC9^2");
// GroupNames label

G:=SmallGroup(243,31);
// by ID

G=gap.SmallGroup(243,31);
# by ID

G:=PCGroup([5,-3,3,3,-3,3,135,276]);
// Polycyclic

G:=Group<a,b,c|a^3=b^9=c^9=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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