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G = C10×C30order 300 = 22·3·52

Abelian group of type [10,30]

direct product, abelian, monomial

Aliases: C10×C30, SmallGroup(300,49)

Series: Derived Chief Lower central Upper central

C1 — C10×C30
C1C5C52C5×C15C5×C30 — C10×C30
C1 — C10×C30
C1 — C10×C30

Generators and relations for C10×C30
 G = < a,b | a10=b30=1, ab=ba >

Subgroups: 80, all normal (8 characteristic)
C1, C2 [×3], C3, C22, C5 [×6], C6 [×3], C10 [×18], C2×C6, C15 [×6], C2×C10 [×6], C52, C30 [×18], C5×C10 [×3], C2×C30 [×6], C5×C15, C102, C5×C30 [×3], C10×C30
Quotients: C1, C2 [×3], C3, C22, C5 [×6], C6 [×3], C10 [×18], C2×C6, C15 [×6], C2×C10 [×6], C52, C30 [×18], C5×C10 [×3], C2×C30 [×6], C5×C15, C102, C5×C30 [×3], C10×C30

Smallest permutation representation of C10×C30
Regular action on 300 points
Generators in S300
(1 61 260 125 101 223 55 204 169 271)(2 62 261 126 102 224 56 205 170 272)(3 63 262 127 103 225 57 206 171 273)(4 64 263 128 104 226 58 207 172 274)(5 65 264 129 105 227 59 208 173 275)(6 66 265 130 106 228 60 209 174 276)(7 67 266 131 107 229 31 210 175 277)(8 68 267 132 108 230 32 181 176 278)(9 69 268 133 109 231 33 182 177 279)(10 70 269 134 110 232 34 183 178 280)(11 71 270 135 111 233 35 184 179 281)(12 72 241 136 112 234 36 185 180 282)(13 73 242 137 113 235 37 186 151 283)(14 74 243 138 114 236 38 187 152 284)(15 75 244 139 115 237 39 188 153 285)(16 76 245 140 116 238 40 189 154 286)(17 77 246 141 117 239 41 190 155 287)(18 78 247 142 118 240 42 191 156 288)(19 79 248 143 119 211 43 192 157 289)(20 80 249 144 120 212 44 193 158 290)(21 81 250 145 91 213 45 194 159 291)(22 82 251 146 92 214 46 195 160 292)(23 83 252 147 93 215 47 196 161 293)(24 84 253 148 94 216 48 197 162 294)(25 85 254 149 95 217 49 198 163 295)(26 86 255 150 96 218 50 199 164 296)(27 87 256 121 97 219 51 200 165 297)(28 88 257 122 98 220 52 201 166 298)(29 89 258 123 99 221 53 202 167 299)(30 90 259 124 100 222 54 203 168 300)
(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)(271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300)

G:=sub<Sym(300)| (1,61,260,125,101,223,55,204,169,271)(2,62,261,126,102,224,56,205,170,272)(3,63,262,127,103,225,57,206,171,273)(4,64,263,128,104,226,58,207,172,274)(5,65,264,129,105,227,59,208,173,275)(6,66,265,130,106,228,60,209,174,276)(7,67,266,131,107,229,31,210,175,277)(8,68,267,132,108,230,32,181,176,278)(9,69,268,133,109,231,33,182,177,279)(10,70,269,134,110,232,34,183,178,280)(11,71,270,135,111,233,35,184,179,281)(12,72,241,136,112,234,36,185,180,282)(13,73,242,137,113,235,37,186,151,283)(14,74,243,138,114,236,38,187,152,284)(15,75,244,139,115,237,39,188,153,285)(16,76,245,140,116,238,40,189,154,286)(17,77,246,141,117,239,41,190,155,287)(18,78,247,142,118,240,42,191,156,288)(19,79,248,143,119,211,43,192,157,289)(20,80,249,144,120,212,44,193,158,290)(21,81,250,145,91,213,45,194,159,291)(22,82,251,146,92,214,46,195,160,292)(23,83,252,147,93,215,47,196,161,293)(24,84,253,148,94,216,48,197,162,294)(25,85,254,149,95,217,49,198,163,295)(26,86,255,150,96,218,50,199,164,296)(27,87,256,121,97,219,51,200,165,297)(28,88,257,122,98,220,52,201,166,298)(29,89,258,123,99,221,53,202,167,299)(30,90,259,124,100,222,54,203,168,300), (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)(271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300)>;

G:=Group( (1,61,260,125,101,223,55,204,169,271)(2,62,261,126,102,224,56,205,170,272)(3,63,262,127,103,225,57,206,171,273)(4,64,263,128,104,226,58,207,172,274)(5,65,264,129,105,227,59,208,173,275)(6,66,265,130,106,228,60,209,174,276)(7,67,266,131,107,229,31,210,175,277)(8,68,267,132,108,230,32,181,176,278)(9,69,268,133,109,231,33,182,177,279)(10,70,269,134,110,232,34,183,178,280)(11,71,270,135,111,233,35,184,179,281)(12,72,241,136,112,234,36,185,180,282)(13,73,242,137,113,235,37,186,151,283)(14,74,243,138,114,236,38,187,152,284)(15,75,244,139,115,237,39,188,153,285)(16,76,245,140,116,238,40,189,154,286)(17,77,246,141,117,239,41,190,155,287)(18,78,247,142,118,240,42,191,156,288)(19,79,248,143,119,211,43,192,157,289)(20,80,249,144,120,212,44,193,158,290)(21,81,250,145,91,213,45,194,159,291)(22,82,251,146,92,214,46,195,160,292)(23,83,252,147,93,215,47,196,161,293)(24,84,253,148,94,216,48,197,162,294)(25,85,254,149,95,217,49,198,163,295)(26,86,255,150,96,218,50,199,164,296)(27,87,256,121,97,219,51,200,165,297)(28,88,257,122,98,220,52,201,166,298)(29,89,258,123,99,221,53,202,167,299)(30,90,259,124,100,222,54,203,168,300), (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)(271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300) );

G=PermutationGroup([(1,61,260,125,101,223,55,204,169,271),(2,62,261,126,102,224,56,205,170,272),(3,63,262,127,103,225,57,206,171,273),(4,64,263,128,104,226,58,207,172,274),(5,65,264,129,105,227,59,208,173,275),(6,66,265,130,106,228,60,209,174,276),(7,67,266,131,107,229,31,210,175,277),(8,68,267,132,108,230,32,181,176,278),(9,69,268,133,109,231,33,182,177,279),(10,70,269,134,110,232,34,183,178,280),(11,71,270,135,111,233,35,184,179,281),(12,72,241,136,112,234,36,185,180,282),(13,73,242,137,113,235,37,186,151,283),(14,74,243,138,114,236,38,187,152,284),(15,75,244,139,115,237,39,188,153,285),(16,76,245,140,116,238,40,189,154,286),(17,77,246,141,117,239,41,190,155,287),(18,78,247,142,118,240,42,191,156,288),(19,79,248,143,119,211,43,192,157,289),(20,80,249,144,120,212,44,193,158,290),(21,81,250,145,91,213,45,194,159,291),(22,82,251,146,92,214,46,195,160,292),(23,83,252,147,93,215,47,196,161,293),(24,84,253,148,94,216,48,197,162,294),(25,85,254,149,95,217,49,198,163,295),(26,86,255,150,96,218,50,199,164,296),(27,87,256,121,97,219,51,200,165,297),(28,88,257,122,98,220,52,201,166,298),(29,89,258,123,99,221,53,202,167,299),(30,90,259,124,100,222,54,203,168,300)], [(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),(271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300)])

300 conjugacy classes

class 1 2A2B2C3A3B5A···5X6A···6F10A···10BT15A···15AV30A···30EN
order1222335···56···610···1015···1530···30
size1111111···11···11···11···11···1

300 irreducible representations

dim11111111
type++
imageC1C2C3C5C6C10C15C30
kernelC10×C30C5×C30C102C2×C30C5×C10C30C2×C10C10
# reps1322467248144

Matrix representation of C10×C30 in GL2(𝔽31) generated by

20
027
,
230
013
G:=sub<GL(2,GF(31))| [2,0,0,27],[23,0,0,13] >;

C10×C30 in GAP, Magma, Sage, TeX

C_{10}\times C_{30}
% in TeX

G:=Group("C10xC30");
// GroupNames label

G:=SmallGroup(300,49);
// by ID

G=gap.SmallGroup(300,49);
# by ID

G:=PCGroup([5,-2,-2,-3,-5,-5]);
// Polycyclic

G:=Group<a,b|a^10=b^30=1,a*b=b*a>;
// generators/relations

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