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G = C6×C48order 288 = 25·32

Abelian group of type [6,48]

direct product, abelian, monomial

Aliases: C6×C48, SmallGroup(288,327)

Series: Derived Chief Lower central Upper central

C1 — C6×C48
C1C2C4C8C24C3×C24C3×C48 — C6×C48
C1 — C6×C48
C1 — C6×C48

Generators and relations for C6×C48
 G = < a,b | a6=b48=1, ab=ba >

Subgroups: 84, all normal (20 characteristic)
C1, C2, C2 [×2], C3 [×4], C4 [×2], C22, C6 [×12], C8 [×2], C2×C4, C32, C12 [×8], C2×C6 [×4], C16 [×2], C2×C8, C3×C6, C3×C6 [×2], C24 [×8], C2×C12 [×4], C2×C16, C3×C12 [×2], C62, C48 [×8], C2×C24 [×4], C3×C24 [×2], C6×C12, C2×C48 [×4], C3×C48 [×2], C6×C24, C6×C48
Quotients: C1, C2 [×3], C3 [×4], C4 [×2], C22, C6 [×12], C8 [×2], C2×C4, C32, C12 [×8], C2×C6 [×4], C16 [×2], C2×C8, C3×C6 [×3], C24 [×8], C2×C12 [×4], C2×C16, C3×C12 [×2], C62, C48 [×8], C2×C24 [×4], C3×C24 [×2], C6×C12, C2×C48 [×4], C3×C48 [×2], C6×C24, C6×C48

Smallest permutation representation of C6×C48
Regular action on 288 points
Generators in S288
(1 248 133 87 186 226)(2 249 134 88 187 227)(3 250 135 89 188 228)(4 251 136 90 189 229)(5 252 137 91 190 230)(6 253 138 92 191 231)(7 254 139 93 192 232)(8 255 140 94 145 233)(9 256 141 95 146 234)(10 257 142 96 147 235)(11 258 143 49 148 236)(12 259 144 50 149 237)(13 260 97 51 150 238)(14 261 98 52 151 239)(15 262 99 53 152 240)(16 263 100 54 153 193)(17 264 101 55 154 194)(18 265 102 56 155 195)(19 266 103 57 156 196)(20 267 104 58 157 197)(21 268 105 59 158 198)(22 269 106 60 159 199)(23 270 107 61 160 200)(24 271 108 62 161 201)(25 272 109 63 162 202)(26 273 110 64 163 203)(27 274 111 65 164 204)(28 275 112 66 165 205)(29 276 113 67 166 206)(30 277 114 68 167 207)(31 278 115 69 168 208)(32 279 116 70 169 209)(33 280 117 71 170 210)(34 281 118 72 171 211)(35 282 119 73 172 212)(36 283 120 74 173 213)(37 284 121 75 174 214)(38 285 122 76 175 215)(39 286 123 77 176 216)(40 287 124 78 177 217)(41 288 125 79 178 218)(42 241 126 80 179 219)(43 242 127 81 180 220)(44 243 128 82 181 221)(45 244 129 83 182 222)(46 245 130 84 183 223)(47 246 131 85 184 224)(48 247 132 86 185 225)
(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)

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

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

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

288 conjugacy classes

class 1 2A2B2C3A···3H4A4B4C4D6A···6X8A···8H12A···12AF16A···16P24A···24BL48A···48DX
order12223···344446···68···812···1216···1624···2448···48
size11111···111111···11···11···11···11···11···1

288 irreducible representations

dim1111111111111111
type+++
imageC1C2C2C3C4C4C6C6C8C8C12C12C16C24C24C48
kernelC6×C48C3×C48C6×C24C2×C48C3×C24C6×C12C48C2×C24C3×C12C62C24C2×C12C3×C6C12C2×C6C6
# reps121822168441616163232128

Matrix representation of C6×C48 in GL2(𝔽97) generated by

360
062
,
850
061
G:=sub<GL(2,GF(97))| [36,0,0,62],[85,0,0,61] >;

C6×C48 in GAP, Magma, Sage, TeX

C_6\times C_{48}
% in TeX

G:=Group("C6xC48");
// GroupNames label

G:=SmallGroup(288,327);
// by ID

G=gap.SmallGroup(288,327);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-2,252,102,124]);
// Polycyclic

G:=Group<a,b|a^6=b^48=1,a*b=b*a>;
// generators/relations

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