direct product, abelian, monomial, 3-elementary
Aliases: C32×C30, SmallGroup(270,30)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C32×C30 |
| C1 — C32×C30 |
| C1 — C32×C30 |
Generators and relations for C32×C30
G = < a,b,c | a3=b3=c30=1, ab=ba, ac=ca, bc=cb >
Subgroups: 112, all normal (8 characteristic)
C1, C2, C3, C5, C6, C32, C10, C15, C3×C6, C33, C30, C3×C15, C32×C6, C3×C30, C32×C15, C32×C30
Quotients: C1, C2, C3, C5, C6, C32, C10, C15, C3×C6, C33, C30, C3×C15, C32×C6, C3×C30, C32×C15, C32×C30
►Regular action on 270 points
Generators in S270
(1 82 147)(2 83 148)(3 84 149)(4 85 150)(5 86 121)(6 87 122)(7 88 123)(8 89 124)(9 90 125)(10 61 126)(11 62 127)(12 63 128)(13 64 129)(14 65 130)(15 66 131)(16 67 132)(17 68 133)(18 69 134)(19 70 135)(20 71 136)(21 72 137)(22 73 138)(23 74 139)(24 75 140)(25 76 141)(26 77 142)(27 78 143)(28 79 144)(29 80 145)(30 81 146)(31 111 244)(32 112 245)(33 113 246)(34 114 247)(35 115 248)(36 116 249)(37 117 250)(38 118 251)(39 119 252)(40 120 253)(41 91 254)(42 92 255)(43 93 256)(44 94 257)(45 95 258)(46 96 259)(47 97 260)(48 98 261)(49 99 262)(50 100 263)(51 101 264)(52 102 265)(53 103 266)(54 104 267)(55 105 268)(56 106 269)(57 107 270)(58 108 241)(59 109 242)(60 110 243)(151 225 200)(152 226 201)(153 227 202)(154 228 203)(155 229 204)(156 230 205)(157 231 206)(158 232 207)(159 233 208)(160 234 209)(161 235 210)(162 236 181)(163 237 182)(164 238 183)(165 239 184)(166 240 185)(167 211 186)(168 212 187)(169 213 188)(170 214 189)(171 215 190)(172 216 191)(173 217 192)(174 218 193)(175 219 194)(176 220 195)(177 221 196)(178 222 197)(179 223 198)(180 224 199)
(1 34 181)(2 35 182)(3 36 183)(4 37 184)(5 38 185)(6 39 186)(7 40 187)(8 41 188)(9 42 189)(10 43 190)(11 44 191)(12 45 192)(13 46 193)(14 47 194)(15 48 195)(16 49 196)(17 50 197)(18 51 198)(19 52 199)(20 53 200)(21 54 201)(22 55 202)(23 56 203)(24 57 204)(25 58 205)(26 59 206)(27 60 207)(28 31 208)(29 32 209)(30 33 210)(61 93 171)(62 94 172)(63 95 173)(64 96 174)(65 97 175)(66 98 176)(67 99 177)(68 100 178)(69 101 179)(70 102 180)(71 103 151)(72 104 152)(73 105 153)(74 106 154)(75 107 155)(76 108 156)(77 109 157)(78 110 158)(79 111 159)(80 112 160)(81 113 161)(82 114 162)(83 115 163)(84 116 164)(85 117 165)(86 118 166)(87 119 167)(88 120 168)(89 91 169)(90 92 170)(121 251 240)(122 252 211)(123 253 212)(124 254 213)(125 255 214)(126 256 215)(127 257 216)(128 258 217)(129 259 218)(130 260 219)(131 261 220)(132 262 221)(133 263 222)(134 264 223)(135 265 224)(136 266 225)(137 267 226)(138 268 227)(139 269 228)(140 270 229)(141 241 230)(142 242 231)(143 243 232)(144 244 233)(145 245 234)(146 246 235)(147 247 236)(148 248 237)(149 249 238)(150 250 239)
(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)(241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270)
G:=sub<Sym(270)| (1,82,147)(2,83,148)(3,84,149)(4,85,150)(5,86,121)(6,87,122)(7,88,123)(8,89,124)(9,90,125)(10,61,126)(11,62,127)(12,63,128)(13,64,129)(14,65,130)(15,66,131)(16,67,132)(17,68,133)(18,69,134)(19,70,135)(20,71,136)(21,72,137)(22,73,138)(23,74,139)(24,75,140)(25,76,141)(26,77,142)(27,78,143)(28,79,144)(29,80,145)(30,81,146)(31,111,244)(32,112,245)(33,113,246)(34,114,247)(35,115,248)(36,116,249)(37,117,250)(38,118,251)(39,119,252)(40,120,253)(41,91,254)(42,92,255)(43,93,256)(44,94,257)(45,95,258)(46,96,259)(47,97,260)(48,98,261)(49,99,262)(50,100,263)(51,101,264)(52,102,265)(53,103,266)(54,104,267)(55,105,268)(56,106,269)(57,107,270)(58,108,241)(59,109,242)(60,110,243)(151,225,200)(152,226,201)(153,227,202)(154,228,203)(155,229,204)(156,230,205)(157,231,206)(158,232,207)(159,233,208)(160,234,209)(161,235,210)(162,236,181)(163,237,182)(164,238,183)(165,239,184)(166,240,185)(167,211,186)(168,212,187)(169,213,188)(170,214,189)(171,215,190)(172,216,191)(173,217,192)(174,218,193)(175,219,194)(176,220,195)(177,221,196)(178,222,197)(179,223,198)(180,224,199), (1,34,181)(2,35,182)(3,36,183)(4,37,184)(5,38,185)(6,39,186)(7,40,187)(8,41,188)(9,42,189)(10,43,190)(11,44,191)(12,45,192)(13,46,193)(14,47,194)(15,48,195)(16,49,196)(17,50,197)(18,51,198)(19,52,199)(20,53,200)(21,54,201)(22,55,202)(23,56,203)(24,57,204)(25,58,205)(26,59,206)(27,60,207)(28,31,208)(29,32,209)(30,33,210)(61,93,171)(62,94,172)(63,95,173)(64,96,174)(65,97,175)(66,98,176)(67,99,177)(68,100,178)(69,101,179)(70,102,180)(71,103,151)(72,104,152)(73,105,153)(74,106,154)(75,107,155)(76,108,156)(77,109,157)(78,110,158)(79,111,159)(80,112,160)(81,113,161)(82,114,162)(83,115,163)(84,116,164)(85,117,165)(86,118,166)(87,119,167)(88,120,168)(89,91,169)(90,92,170)(121,251,240)(122,252,211)(123,253,212)(124,254,213)(125,255,214)(126,256,215)(127,257,216)(128,258,217)(129,259,218)(130,260,219)(131,261,220)(132,262,221)(133,263,222)(134,264,223)(135,265,224)(136,266,225)(137,267,226)(138,268,227)(139,269,228)(140,270,229)(141,241,230)(142,242,231)(143,243,232)(144,244,233)(145,245,234)(146,246,235)(147,247,236)(148,248,237)(149,249,238)(150,250,239), (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)(241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270)>;
G:=Group( (1,82,147)(2,83,148)(3,84,149)(4,85,150)(5,86,121)(6,87,122)(7,88,123)(8,89,124)(9,90,125)(10,61,126)(11,62,127)(12,63,128)(13,64,129)(14,65,130)(15,66,131)(16,67,132)(17,68,133)(18,69,134)(19,70,135)(20,71,136)(21,72,137)(22,73,138)(23,74,139)(24,75,140)(25,76,141)(26,77,142)(27,78,143)(28,79,144)(29,80,145)(30,81,146)(31,111,244)(32,112,245)(33,113,246)(34,114,247)(35,115,248)(36,116,249)(37,117,250)(38,118,251)(39,119,252)(40,120,253)(41,91,254)(42,92,255)(43,93,256)(44,94,257)(45,95,258)(46,96,259)(47,97,260)(48,98,261)(49,99,262)(50,100,263)(51,101,264)(52,102,265)(53,103,266)(54,104,267)(55,105,268)(56,106,269)(57,107,270)(58,108,241)(59,109,242)(60,110,243)(151,225,200)(152,226,201)(153,227,202)(154,228,203)(155,229,204)(156,230,205)(157,231,206)(158,232,207)(159,233,208)(160,234,209)(161,235,210)(162,236,181)(163,237,182)(164,238,183)(165,239,184)(166,240,185)(167,211,186)(168,212,187)(169,213,188)(170,214,189)(171,215,190)(172,216,191)(173,217,192)(174,218,193)(175,219,194)(176,220,195)(177,221,196)(178,222,197)(179,223,198)(180,224,199), (1,34,181)(2,35,182)(3,36,183)(4,37,184)(5,38,185)(6,39,186)(7,40,187)(8,41,188)(9,42,189)(10,43,190)(11,44,191)(12,45,192)(13,46,193)(14,47,194)(15,48,195)(16,49,196)(17,50,197)(18,51,198)(19,52,199)(20,53,200)(21,54,201)(22,55,202)(23,56,203)(24,57,204)(25,58,205)(26,59,206)(27,60,207)(28,31,208)(29,32,209)(30,33,210)(61,93,171)(62,94,172)(63,95,173)(64,96,174)(65,97,175)(66,98,176)(67,99,177)(68,100,178)(69,101,179)(70,102,180)(71,103,151)(72,104,152)(73,105,153)(74,106,154)(75,107,155)(76,108,156)(77,109,157)(78,110,158)(79,111,159)(80,112,160)(81,113,161)(82,114,162)(83,115,163)(84,116,164)(85,117,165)(86,118,166)(87,119,167)(88,120,168)(89,91,169)(90,92,170)(121,251,240)(122,252,211)(123,253,212)(124,254,213)(125,255,214)(126,256,215)(127,257,216)(128,258,217)(129,259,218)(130,260,219)(131,261,220)(132,262,221)(133,263,222)(134,264,223)(135,265,224)(136,266,225)(137,267,226)(138,268,227)(139,269,228)(140,270,229)(141,241,230)(142,242,231)(143,243,232)(144,244,233)(145,245,234)(146,246,235)(147,247,236)(148,248,237)(149,249,238)(150,250,239), (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)(241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270) );
G=PermutationGroup([[(1,82,147),(2,83,148),(3,84,149),(4,85,150),(5,86,121),(6,87,122),(7,88,123),(8,89,124),(9,90,125),(10,61,126),(11,62,127),(12,63,128),(13,64,129),(14,65,130),(15,66,131),(16,67,132),(17,68,133),(18,69,134),(19,70,135),(20,71,136),(21,72,137),(22,73,138),(23,74,139),(24,75,140),(25,76,141),(26,77,142),(27,78,143),(28,79,144),(29,80,145),(30,81,146),(31,111,244),(32,112,245),(33,113,246),(34,114,247),(35,115,248),(36,116,249),(37,117,250),(38,118,251),(39,119,252),(40,120,253),(41,91,254),(42,92,255),(43,93,256),(44,94,257),(45,95,258),(46,96,259),(47,97,260),(48,98,261),(49,99,262),(50,100,263),(51,101,264),(52,102,265),(53,103,266),(54,104,267),(55,105,268),(56,106,269),(57,107,270),(58,108,241),(59,109,242),(60,110,243),(151,225,200),(152,226,201),(153,227,202),(154,228,203),(155,229,204),(156,230,205),(157,231,206),(158,232,207),(159,233,208),(160,234,209),(161,235,210),(162,236,181),(163,237,182),(164,238,183),(165,239,184),(166,240,185),(167,211,186),(168,212,187),(169,213,188),(170,214,189),(171,215,190),(172,216,191),(173,217,192),(174,218,193),(175,219,194),(176,220,195),(177,221,196),(178,222,197),(179,223,198),(180,224,199)], [(1,34,181),(2,35,182),(3,36,183),(4,37,184),(5,38,185),(6,39,186),(7,40,187),(8,41,188),(9,42,189),(10,43,190),(11,44,191),(12,45,192),(13,46,193),(14,47,194),(15,48,195),(16,49,196),(17,50,197),(18,51,198),(19,52,199),(20,53,200),(21,54,201),(22,55,202),(23,56,203),(24,57,204),(25,58,205),(26,59,206),(27,60,207),(28,31,208),(29,32,209),(30,33,210),(61,93,171),(62,94,172),(63,95,173),(64,96,174),(65,97,175),(66,98,176),(67,99,177),(68,100,178),(69,101,179),(70,102,180),(71,103,151),(72,104,152),(73,105,153),(74,106,154),(75,107,155),(76,108,156),(77,109,157),(78,110,158),(79,111,159),(80,112,160),(81,113,161),(82,114,162),(83,115,163),(84,116,164),(85,117,165),(86,118,166),(87,119,167),(88,120,168),(89,91,169),(90,92,170),(121,251,240),(122,252,211),(123,253,212),(124,254,213),(125,255,214),(126,256,215),(127,257,216),(128,258,217),(129,259,218),(130,260,219),(131,261,220),(132,262,221),(133,263,222),(134,264,223),(135,265,224),(136,266,225),(137,267,226),(138,268,227),(139,269,228),(140,270,229),(141,241,230),(142,242,231),(143,243,232),(144,244,233),(145,245,234),(146,246,235),(147,247,236),(148,248,237),(149,249,238),(150,250,239)], [(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),(241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270)]])
270 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3Z | 5A | 5B | 5C | 5D | 6A | ··· | 6Z | 10A | 10B | 10C | 10D | 15A | ··· | 15CZ | 30A | ··· | 30CZ |
| order | 1 | 2 | 3 | ··· | 3 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 10 | 10 | 10 | 10 | 15 | ··· | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
270 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C5 | C6 | C10 | C15 | C30 |
| kernel | C32×C30 | C32×C15 | C3×C30 | C32×C6 | C3×C15 | C33 | C3×C6 | C32 |
| # reps | 1 | 1 | 26 | 4 | 26 | 4 | 104 | 104 |
Matrix representation of C32×C30 ►in GL3(𝔽31) generated by
| 25 | 0 | 0 |
| 0 | 5 | 0 |
| 0 | 0 | 25 |
| 25 | 0 | 0 |
| 0 | 25 | 0 |
| 0 | 0 | 25 |
| 22 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 0 | 3 |
G:=sub<GL(3,GF(31))| [25,0,0,0,5,0,0,0,25],[25,0,0,0,25,0,0,0,25],[22,0,0,0,9,0,0,0,3] >;
C32×C30 in GAP, Magma, Sage, TeX
C_3^2\times C_{30} % in TeX
G:=Group("C3^2xC30"); // GroupNames label
G:=SmallGroup(270,30);
// by ID
G=gap.SmallGroup(270,30);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-5]);
// Polycyclic
G:=Group<a,b,c|a^3=b^3=c^30=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations