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G = C32×C30order 270 = 2·33·5

Abelian group of type [3,3,30]

direct product, abelian, monomial, 3-elementary

Aliases: C32×C30, SmallGroup(270,30)

Series: Derived Chief Lower central Upper central

C1 — C32×C30
C1C5C15C3×C15C32×C15 — 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 [×13], C5, C6 [×13], C32 [×13], C10, C15 [×13], C3×C6 [×13], C33, C30 [×13], C3×C15 [×13], C32×C6, C3×C30 [×13], C32×C15, C32×C30
Quotients: C1, C2, C3 [×13], C5, C6 [×13], C32 [×13], C10, C15 [×13], C3×C6 [×13], C33, C30 [×13], C3×C15 [×13], C32×C6, C3×C30 [×13], C32×C15, C32×C30

Smallest permutation representation of C32×C30
Regular action on 270 points
Generators in S270
(1 245 96)(2 246 97)(3 247 98)(4 248 99)(5 249 100)(6 250 101)(7 251 102)(8 252 103)(9 253 104)(10 254 105)(11 255 106)(12 256 107)(13 257 108)(14 258 109)(15 259 110)(16 260 111)(17 261 112)(18 262 113)(19 263 114)(20 264 115)(21 265 116)(22 266 117)(23 267 118)(24 268 119)(25 269 120)(26 270 91)(27 241 92)(28 242 93)(29 243 94)(30 244 95)(31 165 226)(32 166 227)(33 167 228)(34 168 229)(35 169 230)(36 170 231)(37 171 232)(38 172 233)(39 173 234)(40 174 235)(41 175 236)(42 176 237)(43 177 238)(44 178 239)(45 179 240)(46 180 211)(47 151 212)(48 152 213)(49 153 214)(50 154 215)(51 155 216)(52 156 217)(53 157 218)(54 158 219)(55 159 220)(56 160 221)(57 161 222)(58 162 223)(59 163 224)(60 164 225)(61 187 122)(62 188 123)(63 189 124)(64 190 125)(65 191 126)(66 192 127)(67 193 128)(68 194 129)(69 195 130)(70 196 131)(71 197 132)(72 198 133)(73 199 134)(74 200 135)(75 201 136)(76 202 137)(77 203 138)(78 204 139)(79 205 140)(80 206 141)(81 207 142)(82 208 143)(83 209 144)(84 210 145)(85 181 146)(86 182 147)(87 183 148)(88 184 149)(89 185 150)(90 186 121)
(1 161 62)(2 162 63)(3 163 64)(4 164 65)(5 165 66)(6 166 67)(7 167 68)(8 168 69)(9 169 70)(10 170 71)(11 171 72)(12 172 73)(13 173 74)(14 174 75)(15 175 76)(16 176 77)(17 177 78)(18 178 79)(19 179 80)(20 180 81)(21 151 82)(22 152 83)(23 153 84)(24 154 85)(25 155 86)(26 156 87)(27 157 88)(28 158 89)(29 159 90)(30 160 61)(31 127 100)(32 128 101)(33 129 102)(34 130 103)(35 131 104)(36 132 105)(37 133 106)(38 134 107)(39 135 108)(40 136 109)(41 137 110)(42 138 111)(43 139 112)(44 140 113)(45 141 114)(46 142 115)(47 143 116)(48 144 117)(49 145 118)(50 146 119)(51 147 120)(52 148 91)(53 149 92)(54 150 93)(55 121 94)(56 122 95)(57 123 96)(58 124 97)(59 125 98)(60 126 99)(181 268 215)(182 269 216)(183 270 217)(184 241 218)(185 242 219)(186 243 220)(187 244 221)(188 245 222)(189 246 223)(190 247 224)(191 248 225)(192 249 226)(193 250 227)(194 251 228)(195 252 229)(196 253 230)(197 254 231)(198 255 232)(199 256 233)(200 257 234)(201 258 235)(202 259 236)(203 260 237)(204 261 238)(205 262 239)(206 263 240)(207 264 211)(208 265 212)(209 266 213)(210 267 214)
(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)| (1,245,96)(2,246,97)(3,247,98)(4,248,99)(5,249,100)(6,250,101)(7,251,102)(8,252,103)(9,253,104)(10,254,105)(11,255,106)(12,256,107)(13,257,108)(14,258,109)(15,259,110)(16,260,111)(17,261,112)(18,262,113)(19,263,114)(20,264,115)(21,265,116)(22,266,117)(23,267,118)(24,268,119)(25,269,120)(26,270,91)(27,241,92)(28,242,93)(29,243,94)(30,244,95)(31,165,226)(32,166,227)(33,167,228)(34,168,229)(35,169,230)(36,170,231)(37,171,232)(38,172,233)(39,173,234)(40,174,235)(41,175,236)(42,176,237)(43,177,238)(44,178,239)(45,179,240)(46,180,211)(47,151,212)(48,152,213)(49,153,214)(50,154,215)(51,155,216)(52,156,217)(53,157,218)(54,158,219)(55,159,220)(56,160,221)(57,161,222)(58,162,223)(59,163,224)(60,164,225)(61,187,122)(62,188,123)(63,189,124)(64,190,125)(65,191,126)(66,192,127)(67,193,128)(68,194,129)(69,195,130)(70,196,131)(71,197,132)(72,198,133)(73,199,134)(74,200,135)(75,201,136)(76,202,137)(77,203,138)(78,204,139)(79,205,140)(80,206,141)(81,207,142)(82,208,143)(83,209,144)(84,210,145)(85,181,146)(86,182,147)(87,183,148)(88,184,149)(89,185,150)(90,186,121), (1,161,62)(2,162,63)(3,163,64)(4,164,65)(5,165,66)(6,166,67)(7,167,68)(8,168,69)(9,169,70)(10,170,71)(11,171,72)(12,172,73)(13,173,74)(14,174,75)(15,175,76)(16,176,77)(17,177,78)(18,178,79)(19,179,80)(20,180,81)(21,151,82)(22,152,83)(23,153,84)(24,154,85)(25,155,86)(26,156,87)(27,157,88)(28,158,89)(29,159,90)(30,160,61)(31,127,100)(32,128,101)(33,129,102)(34,130,103)(35,131,104)(36,132,105)(37,133,106)(38,134,107)(39,135,108)(40,136,109)(41,137,110)(42,138,111)(43,139,112)(44,140,113)(45,141,114)(46,142,115)(47,143,116)(48,144,117)(49,145,118)(50,146,119)(51,147,120)(52,148,91)(53,149,92)(54,150,93)(55,121,94)(56,122,95)(57,123,96)(58,124,97)(59,125,98)(60,126,99)(181,268,215)(182,269,216)(183,270,217)(184,241,218)(185,242,219)(186,243,220)(187,244,221)(188,245,222)(189,246,223)(190,247,224)(191,248,225)(192,249,226)(193,250,227)(194,251,228)(195,252,229)(196,253,230)(197,254,231)(198,255,232)(199,256,233)(200,257,234)(201,258,235)(202,259,236)(203,260,237)(204,261,238)(205,262,239)(206,263,240)(207,264,211)(208,265,212)(209,266,213)(210,267,214), (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( (1,245,96)(2,246,97)(3,247,98)(4,248,99)(5,249,100)(6,250,101)(7,251,102)(8,252,103)(9,253,104)(10,254,105)(11,255,106)(12,256,107)(13,257,108)(14,258,109)(15,259,110)(16,260,111)(17,261,112)(18,262,113)(19,263,114)(20,264,115)(21,265,116)(22,266,117)(23,267,118)(24,268,119)(25,269,120)(26,270,91)(27,241,92)(28,242,93)(29,243,94)(30,244,95)(31,165,226)(32,166,227)(33,167,228)(34,168,229)(35,169,230)(36,170,231)(37,171,232)(38,172,233)(39,173,234)(40,174,235)(41,175,236)(42,176,237)(43,177,238)(44,178,239)(45,179,240)(46,180,211)(47,151,212)(48,152,213)(49,153,214)(50,154,215)(51,155,216)(52,156,217)(53,157,218)(54,158,219)(55,159,220)(56,160,221)(57,161,222)(58,162,223)(59,163,224)(60,164,225)(61,187,122)(62,188,123)(63,189,124)(64,190,125)(65,191,126)(66,192,127)(67,193,128)(68,194,129)(69,195,130)(70,196,131)(71,197,132)(72,198,133)(73,199,134)(74,200,135)(75,201,136)(76,202,137)(77,203,138)(78,204,139)(79,205,140)(80,206,141)(81,207,142)(82,208,143)(83,209,144)(84,210,145)(85,181,146)(86,182,147)(87,183,148)(88,184,149)(89,185,150)(90,186,121), (1,161,62)(2,162,63)(3,163,64)(4,164,65)(5,165,66)(6,166,67)(7,167,68)(8,168,69)(9,169,70)(10,170,71)(11,171,72)(12,172,73)(13,173,74)(14,174,75)(15,175,76)(16,176,77)(17,177,78)(18,178,79)(19,179,80)(20,180,81)(21,151,82)(22,152,83)(23,153,84)(24,154,85)(25,155,86)(26,156,87)(27,157,88)(28,158,89)(29,159,90)(30,160,61)(31,127,100)(32,128,101)(33,129,102)(34,130,103)(35,131,104)(36,132,105)(37,133,106)(38,134,107)(39,135,108)(40,136,109)(41,137,110)(42,138,111)(43,139,112)(44,140,113)(45,141,114)(46,142,115)(47,143,116)(48,144,117)(49,145,118)(50,146,119)(51,147,120)(52,148,91)(53,149,92)(54,150,93)(55,121,94)(56,122,95)(57,123,96)(58,124,97)(59,125,98)(60,126,99)(181,268,215)(182,269,216)(183,270,217)(184,241,218)(185,242,219)(186,243,220)(187,244,221)(188,245,222)(189,246,223)(190,247,224)(191,248,225)(192,249,226)(193,250,227)(194,251,228)(195,252,229)(196,253,230)(197,254,231)(198,255,232)(199,256,233)(200,257,234)(201,258,235)(202,259,236)(203,260,237)(204,261,238)(205,262,239)(206,263,240)(207,264,211)(208,265,212)(209,266,213)(210,267,214), (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([(1,245,96),(2,246,97),(3,247,98),(4,248,99),(5,249,100),(6,250,101),(7,251,102),(8,252,103),(9,253,104),(10,254,105),(11,255,106),(12,256,107),(13,257,108),(14,258,109),(15,259,110),(16,260,111),(17,261,112),(18,262,113),(19,263,114),(20,264,115),(21,265,116),(22,266,117),(23,267,118),(24,268,119),(25,269,120),(26,270,91),(27,241,92),(28,242,93),(29,243,94),(30,244,95),(31,165,226),(32,166,227),(33,167,228),(34,168,229),(35,169,230),(36,170,231),(37,171,232),(38,172,233),(39,173,234),(40,174,235),(41,175,236),(42,176,237),(43,177,238),(44,178,239),(45,179,240),(46,180,211),(47,151,212),(48,152,213),(49,153,214),(50,154,215),(51,155,216),(52,156,217),(53,157,218),(54,158,219),(55,159,220),(56,160,221),(57,161,222),(58,162,223),(59,163,224),(60,164,225),(61,187,122),(62,188,123),(63,189,124),(64,190,125),(65,191,126),(66,192,127),(67,193,128),(68,194,129),(69,195,130),(70,196,131),(71,197,132),(72,198,133),(73,199,134),(74,200,135),(75,201,136),(76,202,137),(77,203,138),(78,204,139),(79,205,140),(80,206,141),(81,207,142),(82,208,143),(83,209,144),(84,210,145),(85,181,146),(86,182,147),(87,183,148),(88,184,149),(89,185,150),(90,186,121)], [(1,161,62),(2,162,63),(3,163,64),(4,164,65),(5,165,66),(6,166,67),(7,167,68),(8,168,69),(9,169,70),(10,170,71),(11,171,72),(12,172,73),(13,173,74),(14,174,75),(15,175,76),(16,176,77),(17,177,78),(18,178,79),(19,179,80),(20,180,81),(21,151,82),(22,152,83),(23,153,84),(24,154,85),(25,155,86),(26,156,87),(27,157,88),(28,158,89),(29,159,90),(30,160,61),(31,127,100),(32,128,101),(33,129,102),(34,130,103),(35,131,104),(36,132,105),(37,133,106),(38,134,107),(39,135,108),(40,136,109),(41,137,110),(42,138,111),(43,139,112),(44,140,113),(45,141,114),(46,142,115),(47,143,116),(48,144,117),(49,145,118),(50,146,119),(51,147,120),(52,148,91),(53,149,92),(54,150,93),(55,121,94),(56,122,95),(57,123,96),(58,124,97),(59,125,98),(60,126,99),(181,268,215),(182,269,216),(183,270,217),(184,241,218),(185,242,219),(186,243,220),(187,244,221),(188,245,222),(189,246,223),(190,247,224),(191,248,225),(192,249,226),(193,250,227),(194,251,228),(195,252,229),(196,253,230),(197,254,231),(198,255,232),(199,256,233),(200,257,234),(201,258,235),(202,259,236),(203,260,237),(204,261,238),(205,262,239),(206,263,240),(207,264,211),(208,265,212),(209,266,213),(210,267,214)], [(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···3Z5A5B5C5D6A···6Z10A10B10C10D15A···15CZ30A···30CZ
order123···355556···61010101015···1530···30
size111···111111···111111···11···1

270 irreducible representations

dim11111111
type++
imageC1C2C3C5C6C10C15C30
kernelC32×C30C32×C15C3×C30C32×C6C3×C15C33C3×C6C32
# reps11264264104104

Matrix representation of C32×C30 in GL3(𝔽31) generated by

2500
050
0025
,
2500
0250
0025
,
2200
090
003
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

׿
×
𝔽