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## G = C32×C27order 243 = 35

### Abelian group of type [3,3,27]

Aliases: C32×C27, SmallGroup(243,48)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C32×C27
 Chief series C1 — C3 — C9 — C3×C9 — C32×C9 — C32×C27
 Lower central C1 — C32×C27
 Upper central C1 — C32×C27
 Jennings C1 — C3 — C3 — C3 — C3 — C3 — C3 — C9 — C9 — C32×C27

Generators and relations for C32×C27
G = < a,b,c | a3=b3=c27=1, ab=ba, ac=ca, bc=cb >

Subgroups: 72, all normal (6 characteristic)
C1, C3, C3 [×12], C9, C9 [×8], C32 [×13], C27 [×9], C3×C9 [×12], C33, C3×C27 [×12], C32×C9, C32×C27
Quotients: C1, C3 [×13], C9 [×9], C32 [×13], C27 [×9], C3×C9 [×12], C33, C3×C27 [×12], C32×C9, C32×C27

Smallest permutation representation of C32×C27
Regular action on 243 points
Generators in S243
(1 57 51)(2 58 52)(3 59 53)(4 60 54)(5 61 28)(6 62 29)(7 63 30)(8 64 31)(9 65 32)(10 66 33)(11 67 34)(12 68 35)(13 69 36)(14 70 37)(15 71 38)(16 72 39)(17 73 40)(18 74 41)(19 75 42)(20 76 43)(21 77 44)(22 78 45)(23 79 46)(24 80 47)(25 81 48)(26 55 49)(27 56 50)(82 124 237)(83 125 238)(84 126 239)(85 127 240)(86 128 241)(87 129 242)(88 130 243)(89 131 217)(90 132 218)(91 133 219)(92 134 220)(93 135 221)(94 109 222)(95 110 223)(96 111 224)(97 112 225)(98 113 226)(99 114 227)(100 115 228)(101 116 229)(102 117 230)(103 118 231)(104 119 232)(105 120 233)(106 121 234)(107 122 235)(108 123 236)(136 177 198)(137 178 199)(138 179 200)(139 180 201)(140 181 202)(141 182 203)(142 183 204)(143 184 205)(144 185 206)(145 186 207)(146 187 208)(147 188 209)(148 189 210)(149 163 211)(150 164 212)(151 165 213)(152 166 214)(153 167 215)(154 168 216)(155 169 190)(156 170 191)(157 171 192)(158 172 193)(159 173 194)(160 174 195)(161 175 196)(162 176 197)
(1 242 174)(2 243 175)(3 217 176)(4 218 177)(5 219 178)(6 220 179)(7 221 180)(8 222 181)(9 223 182)(10 224 183)(11 225 184)(12 226 185)(13 227 186)(14 228 187)(15 229 188)(16 230 189)(17 231 163)(18 232 164)(19 233 165)(20 234 166)(21 235 167)(22 236 168)(23 237 169)(24 238 170)(25 239 171)(26 240 172)(27 241 173)(28 133 137)(29 134 138)(30 135 139)(31 109 140)(32 110 141)(33 111 142)(34 112 143)(35 113 144)(36 114 145)(37 115 146)(38 116 147)(39 117 148)(40 118 149)(41 119 150)(42 120 151)(43 121 152)(44 122 153)(45 123 154)(46 124 155)(47 125 156)(48 126 157)(49 127 158)(50 128 159)(51 129 160)(52 130 161)(53 131 162)(54 132 136)(55 85 193)(56 86 194)(57 87 195)(58 88 196)(59 89 197)(60 90 198)(61 91 199)(62 92 200)(63 93 201)(64 94 202)(65 95 203)(66 96 204)(67 97 205)(68 98 206)(69 99 207)(70 100 208)(71 101 209)(72 102 210)(73 103 211)(74 104 212)(75 105 213)(76 106 214)(77 107 215)(78 108 216)(79 82 190)(80 83 191)(81 84 192)
(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)

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

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

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

C32×C27 is a maximal subgroup of   C324D27

243 conjugacy classes

 class 1 3A ··· 3Z 9A ··· 9BB 27A ··· 27FF order 1 3 ··· 3 9 ··· 9 27 ··· 27 size 1 1 ··· 1 1 ··· 1 1 ··· 1

243 irreducible representations

 dim 1 1 1 1 1 1 type + image C1 C3 C3 C9 C9 C27 kernel C32×C27 C3×C27 C32×C9 C3×C9 C33 C32 # reps 1 24 2 48 6 162

Matrix representation of C32×C27 in GL3(𝔽109) generated by

 45 0 0 0 1 0 0 0 63
,
 63 0 0 0 1 0 0 0 63
,
 81 0 0 0 25 0 0 0 35
G:=sub<GL(3,GF(109))| [45,0,0,0,1,0,0,0,63],[63,0,0,0,1,0,0,0,63],[81,0,0,0,25,0,0,0,35] >;

C32×C27 in GAP, Magma, Sage, TeX

C_3^2\times C_{27}
% in TeX

G:=Group("C3^2xC27");
// GroupNames label

G:=SmallGroup(243,48);
// by ID

G=gap.SmallGroup(243,48);
# by ID

G:=PCGroup([5,-3,3,3,-3,-3,135,78]);
// Polycyclic

G:=Group<a,b,c|a^3=b^3=c^27=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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