direct product, p-group, abelian, monomial
Aliases: C32×C27, SmallGroup(243,48)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C32×C27 |
| C1 — C32×C27 |
| C1 — 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, C9, C9, C32, C27, C3×C9, C33, C3×C27, C32×C9, C32×C27
Quotients: C1, C3, C9, C32, C27, C3×C9, C33, C3×C27, C32×C9, C32×C27
►Regular action on 243 points
Generators in S243
(1 56 103)(2 57 104)(3 58 105)(4 59 106)(5 60 107)(6 61 108)(7 62 82)(8 63 83)(9 64 84)(10 65 85)(11 66 86)(12 67 87)(13 68 88)(14 69 89)(15 70 90)(16 71 91)(17 72 92)(18 73 93)(19 74 94)(20 75 95)(21 76 96)(22 77 97)(23 78 98)(24 79 99)(25 80 100)(26 81 101)(27 55 102)(28 221 121)(29 222 122)(30 223 123)(31 224 124)(32 225 125)(33 226 126)(34 227 127)(35 228 128)(36 229 129)(37 230 130)(38 231 131)(39 232 132)(40 233 133)(41 234 134)(42 235 135)(43 236 109)(44 237 110)(45 238 111)(46 239 112)(47 240 113)(48 241 114)(49 242 115)(50 243 116)(51 217 117)(52 218 118)(53 219 119)(54 220 120)(136 185 214)(137 186 215)(138 187 216)(139 188 190)(140 189 191)(141 163 192)(142 164 193)(143 165 194)(144 166 195)(145 167 196)(146 168 197)(147 169 198)(148 170 199)(149 171 200)(150 172 201)(151 173 202)(152 174 203)(153 175 204)(154 176 205)(155 177 206)(156 178 207)(157 179 208)(158 180 209)(159 181 210)(160 182 211)(161 183 212)(162 184 213)
(1 196 225)(2 197 226)(3 198 227)(4 199 228)(5 200 229)(6 201 230)(7 202 231)(8 203 232)(9 204 233)(10 205 234)(11 206 235)(12 207 236)(13 208 237)(14 209 238)(15 210 239)(16 211 240)(17 212 241)(18 213 242)(19 214 243)(20 215 217)(21 216 218)(22 190 219)(23 191 220)(24 192 221)(25 193 222)(26 194 223)(27 195 224)(28 99 163)(29 100 164)(30 101 165)(31 102 166)(32 103 167)(33 104 168)(34 105 169)(35 106 170)(36 107 171)(37 108 172)(38 82 173)(39 83 174)(40 84 175)(41 85 176)(42 86 177)(43 87 178)(44 88 179)(45 89 180)(46 90 181)(47 91 182)(48 92 183)(49 93 184)(50 94 185)(51 95 186)(52 96 187)(53 97 188)(54 98 189)(55 144 124)(56 145 125)(57 146 126)(58 147 127)(59 148 128)(60 149 129)(61 150 130)(62 151 131)(63 152 132)(64 153 133)(65 154 134)(66 155 135)(67 156 109)(68 157 110)(69 158 111)(70 159 112)(71 160 113)(72 161 114)(73 162 115)(74 136 116)(75 137 117)(76 138 118)(77 139 119)(78 140 120)(79 141 121)(80 142 122)(81 143 123)
(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)| (1,56,103)(2,57,104)(3,58,105)(4,59,106)(5,60,107)(6,61,108)(7,62,82)(8,63,83)(9,64,84)(10,65,85)(11,66,86)(12,67,87)(13,68,88)(14,69,89)(15,70,90)(16,71,91)(17,72,92)(18,73,93)(19,74,94)(20,75,95)(21,76,96)(22,77,97)(23,78,98)(24,79,99)(25,80,100)(26,81,101)(27,55,102)(28,221,121)(29,222,122)(30,223,123)(31,224,124)(32,225,125)(33,226,126)(34,227,127)(35,228,128)(36,229,129)(37,230,130)(38,231,131)(39,232,132)(40,233,133)(41,234,134)(42,235,135)(43,236,109)(44,237,110)(45,238,111)(46,239,112)(47,240,113)(48,241,114)(49,242,115)(50,243,116)(51,217,117)(52,218,118)(53,219,119)(54,220,120)(136,185,214)(137,186,215)(138,187,216)(139,188,190)(140,189,191)(141,163,192)(142,164,193)(143,165,194)(144,166,195)(145,167,196)(146,168,197)(147,169,198)(148,170,199)(149,171,200)(150,172,201)(151,173,202)(152,174,203)(153,175,204)(154,176,205)(155,177,206)(156,178,207)(157,179,208)(158,180,209)(159,181,210)(160,182,211)(161,183,212)(162,184,213), (1,196,225)(2,197,226)(3,198,227)(4,199,228)(5,200,229)(6,201,230)(7,202,231)(8,203,232)(9,204,233)(10,205,234)(11,206,235)(12,207,236)(13,208,237)(14,209,238)(15,210,239)(16,211,240)(17,212,241)(18,213,242)(19,214,243)(20,215,217)(21,216,218)(22,190,219)(23,191,220)(24,192,221)(25,193,222)(26,194,223)(27,195,224)(28,99,163)(29,100,164)(30,101,165)(31,102,166)(32,103,167)(33,104,168)(34,105,169)(35,106,170)(36,107,171)(37,108,172)(38,82,173)(39,83,174)(40,84,175)(41,85,176)(42,86,177)(43,87,178)(44,88,179)(45,89,180)(46,90,181)(47,91,182)(48,92,183)(49,93,184)(50,94,185)(51,95,186)(52,96,187)(53,97,188)(54,98,189)(55,144,124)(56,145,125)(57,146,126)(58,147,127)(59,148,128)(60,149,129)(61,150,130)(62,151,131)(63,152,132)(64,153,133)(65,154,134)(66,155,135)(67,156,109)(68,157,110)(69,158,111)(70,159,112)(71,160,113)(72,161,114)(73,162,115)(74,136,116)(75,137,117)(76,138,118)(77,139,119)(78,140,120)(79,141,121)(80,142,122)(81,143,123), (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( (1,56,103)(2,57,104)(3,58,105)(4,59,106)(5,60,107)(6,61,108)(7,62,82)(8,63,83)(9,64,84)(10,65,85)(11,66,86)(12,67,87)(13,68,88)(14,69,89)(15,70,90)(16,71,91)(17,72,92)(18,73,93)(19,74,94)(20,75,95)(21,76,96)(22,77,97)(23,78,98)(24,79,99)(25,80,100)(26,81,101)(27,55,102)(28,221,121)(29,222,122)(30,223,123)(31,224,124)(32,225,125)(33,226,126)(34,227,127)(35,228,128)(36,229,129)(37,230,130)(38,231,131)(39,232,132)(40,233,133)(41,234,134)(42,235,135)(43,236,109)(44,237,110)(45,238,111)(46,239,112)(47,240,113)(48,241,114)(49,242,115)(50,243,116)(51,217,117)(52,218,118)(53,219,119)(54,220,120)(136,185,214)(137,186,215)(138,187,216)(139,188,190)(140,189,191)(141,163,192)(142,164,193)(143,165,194)(144,166,195)(145,167,196)(146,168,197)(147,169,198)(148,170,199)(149,171,200)(150,172,201)(151,173,202)(152,174,203)(153,175,204)(154,176,205)(155,177,206)(156,178,207)(157,179,208)(158,180,209)(159,181,210)(160,182,211)(161,183,212)(162,184,213), (1,196,225)(2,197,226)(3,198,227)(4,199,228)(5,200,229)(6,201,230)(7,202,231)(8,203,232)(9,204,233)(10,205,234)(11,206,235)(12,207,236)(13,208,237)(14,209,238)(15,210,239)(16,211,240)(17,212,241)(18,213,242)(19,214,243)(20,215,217)(21,216,218)(22,190,219)(23,191,220)(24,192,221)(25,193,222)(26,194,223)(27,195,224)(28,99,163)(29,100,164)(30,101,165)(31,102,166)(32,103,167)(33,104,168)(34,105,169)(35,106,170)(36,107,171)(37,108,172)(38,82,173)(39,83,174)(40,84,175)(41,85,176)(42,86,177)(43,87,178)(44,88,179)(45,89,180)(46,90,181)(47,91,182)(48,92,183)(49,93,184)(50,94,185)(51,95,186)(52,96,187)(53,97,188)(54,98,189)(55,144,124)(56,145,125)(57,146,126)(58,147,127)(59,148,128)(60,149,129)(61,150,130)(62,151,131)(63,152,132)(64,153,133)(65,154,134)(66,155,135)(67,156,109)(68,157,110)(69,158,111)(70,159,112)(71,160,113)(72,161,114)(73,162,115)(74,136,116)(75,137,117)(76,138,118)(77,139,119)(78,140,120)(79,141,121)(80,142,122)(81,143,123), (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([[(1,56,103),(2,57,104),(3,58,105),(4,59,106),(5,60,107),(6,61,108),(7,62,82),(8,63,83),(9,64,84),(10,65,85),(11,66,86),(12,67,87),(13,68,88),(14,69,89),(15,70,90),(16,71,91),(17,72,92),(18,73,93),(19,74,94),(20,75,95),(21,76,96),(22,77,97),(23,78,98),(24,79,99),(25,80,100),(26,81,101),(27,55,102),(28,221,121),(29,222,122),(30,223,123),(31,224,124),(32,225,125),(33,226,126),(34,227,127),(35,228,128),(36,229,129),(37,230,130),(38,231,131),(39,232,132),(40,233,133),(41,234,134),(42,235,135),(43,236,109),(44,237,110),(45,238,111),(46,239,112),(47,240,113),(48,241,114),(49,242,115),(50,243,116),(51,217,117),(52,218,118),(53,219,119),(54,220,120),(136,185,214),(137,186,215),(138,187,216),(139,188,190),(140,189,191),(141,163,192),(142,164,193),(143,165,194),(144,166,195),(145,167,196),(146,168,197),(147,169,198),(148,170,199),(149,171,200),(150,172,201),(151,173,202),(152,174,203),(153,175,204),(154,176,205),(155,177,206),(156,178,207),(157,179,208),(158,180,209),(159,181,210),(160,182,211),(161,183,212),(162,184,213)], [(1,196,225),(2,197,226),(3,198,227),(4,199,228),(5,200,229),(6,201,230),(7,202,231),(8,203,232),(9,204,233),(10,205,234),(11,206,235),(12,207,236),(13,208,237),(14,209,238),(15,210,239),(16,211,240),(17,212,241),(18,213,242),(19,214,243),(20,215,217),(21,216,218),(22,190,219),(23,191,220),(24,192,221),(25,193,222),(26,194,223),(27,195,224),(28,99,163),(29,100,164),(30,101,165),(31,102,166),(32,103,167),(33,104,168),(34,105,169),(35,106,170),(36,107,171),(37,108,172),(38,82,173),(39,83,174),(40,84,175),(41,85,176),(42,86,177),(43,87,178),(44,88,179),(45,89,180),(46,90,181),(47,91,182),(48,92,183),(49,93,184),(50,94,185),(51,95,186),(52,96,187),(53,97,188),(54,98,189),(55,144,124),(56,145,125),(57,146,126),(58,147,127),(59,148,128),(60,149,129),(61,150,130),(62,151,131),(63,152,132),(64,153,133),(65,154,134),(66,155,135),(67,156,109),(68,157,110),(69,158,111),(70,159,112),(71,160,113),(72,161,114),(73,162,115),(74,136,116),(75,137,117),(76,138,118),(77,139,119),(78,140,120),(79,141,121),(80,142,122),(81,143,123)], [(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
C32⋊4D27
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