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## G = C6×C48order 288 = 25·32

### Abelian group of type [6,48]

Aliases: C6×C48, SmallGroup(288,327)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6×C48
 Chief series C1 — C2 — C4 — C8 — C24 — C3×C24 — C3×C48 — C6×C48
 Lower central C1 — C6×C48
 Upper central 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, C3, C4, C22, C6, C8, C2×C4, C32, C12, C2×C6, C16, C2×C8, C3×C6, C3×C6, C24, C2×C12, C2×C16, C3×C12, C62, C48, C2×C24, C3×C24, C6×C12, C2×C48, C3×C48, C6×C24, C6×C48
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, C32, C12, C2×C6, C16, C2×C8, C3×C6, C24, C2×C12, C2×C16, C3×C12, C62, C48, C2×C24, C3×C24, C6×C12, C2×C48, C3×C48, C6×C24, C6×C48

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

288 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 type + + + image C1 C2 C2 C3 C4 C4 C6 C6 C8 C8 C12 C12 C16 C24 C24 C48 kernel C6×C48 C3×C48 C6×C24 C2×C48 C3×C24 C6×C12 C48 C2×C24 C3×C12 C62 C24 C2×C12 C3×C6 C12 C2×C6 C6 # reps 1 2 1 8 2 2 16 8 4 4 16 16 16 32 32 128

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

 36 0 0 62
,
 85 0 0 61
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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