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G = C2×C6×C24order 288 = 25·32

Abelian group of type [2,6,24]

direct product, abelian, monomial

Aliases: C2×C6×C24, SmallGroup(288,826)

Series: Derived Chief Lower central Upper central

C1 — C2×C6×C24
C1C2C4C12C3×C12C3×C24C6×C24 — C2×C6×C24
C1 — C2×C6×C24
C1 — C2×C6×C24

Generators and relations for C2×C6×C24
 G = < a,b,c | a2=b6=c24=1, ab=ba, ac=ca, bc=cb >

Subgroups: 228, all normal (12 characteristic)
C1, C2, C2 [×6], C3 [×4], C4, C4 [×3], C22 [×7], C6 [×28], C8 [×4], C2×C4 [×6], C23, C32, C12 [×16], C2×C6 [×28], C2×C8 [×6], C22×C4, C3×C6, C3×C6 [×6], C24 [×16], C2×C12 [×24], C22×C6 [×4], C22×C8, C3×C12, C3×C12 [×3], C62 [×7], C2×C24 [×24], C22×C12 [×4], C3×C24 [×4], C6×C12 [×6], C2×C62, C22×C24 [×4], C6×C24 [×6], C2×C6×C12, C2×C6×C24
Quotients: C1, C2 [×7], C3 [×4], C4 [×4], C22 [×7], C6 [×28], C8 [×4], C2×C4 [×6], C23, C32, C12 [×16], C2×C6 [×28], C2×C8 [×6], C22×C4, C3×C6 [×7], C24 [×16], C2×C12 [×24], C22×C6 [×4], C22×C8, C3×C12 [×4], C62 [×7], C2×C24 [×24], C22×C12 [×4], C3×C24 [×4], C6×C12 [×6], C2×C62, C22×C24 [×4], C6×C24 [×6], C2×C6×C12, C2×C6×C24

Smallest permutation representation of C2×C6×C24
Regular action on 288 points
Generators in S288
(1 170)(2 171)(3 172)(4 173)(5 174)(6 175)(7 176)(8 177)(9 178)(10 179)(11 180)(12 181)(13 182)(14 183)(15 184)(16 185)(17 186)(18 187)(19 188)(20 189)(21 190)(22 191)(23 192)(24 169)(25 208)(26 209)(27 210)(28 211)(29 212)(30 213)(31 214)(32 215)(33 216)(34 193)(35 194)(36 195)(37 196)(38 197)(39 198)(40 199)(41 200)(42 201)(43 202)(44 203)(45 204)(46 205)(47 206)(48 207)(49 84)(50 85)(51 86)(52 87)(53 88)(54 89)(55 90)(56 91)(57 92)(58 93)(59 94)(60 95)(61 96)(62 73)(63 74)(64 75)(65 76)(66 77)(67 78)(68 79)(69 80)(70 81)(71 82)(72 83)(97 167)(98 168)(99 145)(100 146)(101 147)(102 148)(103 149)(104 150)(105 151)(106 152)(107 153)(108 154)(109 155)(110 156)(111 157)(112 158)(113 159)(114 160)(115 161)(116 162)(117 163)(118 164)(119 165)(120 166)(121 260)(122 261)(123 262)(124 263)(125 264)(126 241)(127 242)(128 243)(129 244)(130 245)(131 246)(132 247)(133 248)(134 249)(135 250)(136 251)(137 252)(138 253)(139 254)(140 255)(141 256)(142 257)(143 258)(144 259)(217 281)(218 282)(219 283)(220 284)(221 285)(222 286)(223 287)(224 288)(225 265)(226 266)(227 267)(228 268)(229 269)(230 270)(231 271)(232 272)(233 273)(234 274)(235 275)(236 276)(237 277)(238 278)(239 279)(240 280)
(1 210 128 272 81 163)(2 211 129 273 82 164)(3 212 130 274 83 165)(4 213 131 275 84 166)(5 214 132 276 85 167)(6 215 133 277 86 168)(7 216 134 278 87 145)(8 193 135 279 88 146)(9 194 136 280 89 147)(10 195 137 281 90 148)(11 196 138 282 91 149)(12 197 139 283 92 150)(13 198 140 284 93 151)(14 199 141 285 94 152)(15 200 142 286 95 153)(16 201 143 287 96 154)(17 202 144 288 73 155)(18 203 121 265 74 156)(19 204 122 266 75 157)(20 205 123 267 76 158)(21 206 124 268 77 159)(22 207 125 269 78 160)(23 208 126 270 79 161)(24 209 127 271 80 162)(25 241 230 68 115 192)(26 242 231 69 116 169)(27 243 232 70 117 170)(28 244 233 71 118 171)(29 245 234 72 119 172)(30 246 235 49 120 173)(31 247 236 50 97 174)(32 248 237 51 98 175)(33 249 238 52 99 176)(34 250 239 53 100 177)(35 251 240 54 101 178)(36 252 217 55 102 179)(37 253 218 56 103 180)(38 254 219 57 104 181)(39 255 220 58 105 182)(40 256 221 59 106 183)(41 257 222 60 107 184)(42 258 223 61 108 185)(43 259 224 62 109 186)(44 260 225 63 110 187)(45 261 226 64 111 188)(46 262 227 65 112 189)(47 263 228 66 113 190)(48 264 229 67 114 191)
(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)| (1,170)(2,171)(3,172)(4,173)(5,174)(6,175)(7,176)(8,177)(9,178)(10,179)(11,180)(12,181)(13,182)(14,183)(15,184)(16,185)(17,186)(18,187)(19,188)(20,189)(21,190)(22,191)(23,192)(24,169)(25,208)(26,209)(27,210)(28,211)(29,212)(30,213)(31,214)(32,215)(33,216)(34,193)(35,194)(36,195)(37,196)(38,197)(39,198)(40,199)(41,200)(42,201)(43,202)(44,203)(45,204)(46,205)(47,206)(48,207)(49,84)(50,85)(51,86)(52,87)(53,88)(54,89)(55,90)(56,91)(57,92)(58,93)(59,94)(60,95)(61,96)(62,73)(63,74)(64,75)(65,76)(66,77)(67,78)(68,79)(69,80)(70,81)(71,82)(72,83)(97,167)(98,168)(99,145)(100,146)(101,147)(102,148)(103,149)(104,150)(105,151)(106,152)(107,153)(108,154)(109,155)(110,156)(111,157)(112,158)(113,159)(114,160)(115,161)(116,162)(117,163)(118,164)(119,165)(120,166)(121,260)(122,261)(123,262)(124,263)(125,264)(126,241)(127,242)(128,243)(129,244)(130,245)(131,246)(132,247)(133,248)(134,249)(135,250)(136,251)(137,252)(138,253)(139,254)(140,255)(141,256)(142,257)(143,258)(144,259)(217,281)(218,282)(219,283)(220,284)(221,285)(222,286)(223,287)(224,288)(225,265)(226,266)(227,267)(228,268)(229,269)(230,270)(231,271)(232,272)(233,273)(234,274)(235,275)(236,276)(237,277)(238,278)(239,279)(240,280), (1,210,128,272,81,163)(2,211,129,273,82,164)(3,212,130,274,83,165)(4,213,131,275,84,166)(5,214,132,276,85,167)(6,215,133,277,86,168)(7,216,134,278,87,145)(8,193,135,279,88,146)(9,194,136,280,89,147)(10,195,137,281,90,148)(11,196,138,282,91,149)(12,197,139,283,92,150)(13,198,140,284,93,151)(14,199,141,285,94,152)(15,200,142,286,95,153)(16,201,143,287,96,154)(17,202,144,288,73,155)(18,203,121,265,74,156)(19,204,122,266,75,157)(20,205,123,267,76,158)(21,206,124,268,77,159)(22,207,125,269,78,160)(23,208,126,270,79,161)(24,209,127,271,80,162)(25,241,230,68,115,192)(26,242,231,69,116,169)(27,243,232,70,117,170)(28,244,233,71,118,171)(29,245,234,72,119,172)(30,246,235,49,120,173)(31,247,236,50,97,174)(32,248,237,51,98,175)(33,249,238,52,99,176)(34,250,239,53,100,177)(35,251,240,54,101,178)(36,252,217,55,102,179)(37,253,218,56,103,180)(38,254,219,57,104,181)(39,255,220,58,105,182)(40,256,221,59,106,183)(41,257,222,60,107,184)(42,258,223,61,108,185)(43,259,224,62,109,186)(44,260,225,63,110,187)(45,261,226,64,111,188)(46,262,227,65,112,189)(47,263,228,66,113,190)(48,264,229,67,114,191), (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( (1,170)(2,171)(3,172)(4,173)(5,174)(6,175)(7,176)(8,177)(9,178)(10,179)(11,180)(12,181)(13,182)(14,183)(15,184)(16,185)(17,186)(18,187)(19,188)(20,189)(21,190)(22,191)(23,192)(24,169)(25,208)(26,209)(27,210)(28,211)(29,212)(30,213)(31,214)(32,215)(33,216)(34,193)(35,194)(36,195)(37,196)(38,197)(39,198)(40,199)(41,200)(42,201)(43,202)(44,203)(45,204)(46,205)(47,206)(48,207)(49,84)(50,85)(51,86)(52,87)(53,88)(54,89)(55,90)(56,91)(57,92)(58,93)(59,94)(60,95)(61,96)(62,73)(63,74)(64,75)(65,76)(66,77)(67,78)(68,79)(69,80)(70,81)(71,82)(72,83)(97,167)(98,168)(99,145)(100,146)(101,147)(102,148)(103,149)(104,150)(105,151)(106,152)(107,153)(108,154)(109,155)(110,156)(111,157)(112,158)(113,159)(114,160)(115,161)(116,162)(117,163)(118,164)(119,165)(120,166)(121,260)(122,261)(123,262)(124,263)(125,264)(126,241)(127,242)(128,243)(129,244)(130,245)(131,246)(132,247)(133,248)(134,249)(135,250)(136,251)(137,252)(138,253)(139,254)(140,255)(141,256)(142,257)(143,258)(144,259)(217,281)(218,282)(219,283)(220,284)(221,285)(222,286)(223,287)(224,288)(225,265)(226,266)(227,267)(228,268)(229,269)(230,270)(231,271)(232,272)(233,273)(234,274)(235,275)(236,276)(237,277)(238,278)(239,279)(240,280), (1,210,128,272,81,163)(2,211,129,273,82,164)(3,212,130,274,83,165)(4,213,131,275,84,166)(5,214,132,276,85,167)(6,215,133,277,86,168)(7,216,134,278,87,145)(8,193,135,279,88,146)(9,194,136,280,89,147)(10,195,137,281,90,148)(11,196,138,282,91,149)(12,197,139,283,92,150)(13,198,140,284,93,151)(14,199,141,285,94,152)(15,200,142,286,95,153)(16,201,143,287,96,154)(17,202,144,288,73,155)(18,203,121,265,74,156)(19,204,122,266,75,157)(20,205,123,267,76,158)(21,206,124,268,77,159)(22,207,125,269,78,160)(23,208,126,270,79,161)(24,209,127,271,80,162)(25,241,230,68,115,192)(26,242,231,69,116,169)(27,243,232,70,117,170)(28,244,233,71,118,171)(29,245,234,72,119,172)(30,246,235,49,120,173)(31,247,236,50,97,174)(32,248,237,51,98,175)(33,249,238,52,99,176)(34,250,239,53,100,177)(35,251,240,54,101,178)(36,252,217,55,102,179)(37,253,218,56,103,180)(38,254,219,57,104,181)(39,255,220,58,105,182)(40,256,221,59,106,183)(41,257,222,60,107,184)(42,258,223,61,108,185)(43,259,224,62,109,186)(44,260,225,63,110,187)(45,261,226,64,111,188)(46,262,227,65,112,189)(47,263,228,66,113,190)(48,264,229,67,114,191), (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([(1,170),(2,171),(3,172),(4,173),(5,174),(6,175),(7,176),(8,177),(9,178),(10,179),(11,180),(12,181),(13,182),(14,183),(15,184),(16,185),(17,186),(18,187),(19,188),(20,189),(21,190),(22,191),(23,192),(24,169),(25,208),(26,209),(27,210),(28,211),(29,212),(30,213),(31,214),(32,215),(33,216),(34,193),(35,194),(36,195),(37,196),(38,197),(39,198),(40,199),(41,200),(42,201),(43,202),(44,203),(45,204),(46,205),(47,206),(48,207),(49,84),(50,85),(51,86),(52,87),(53,88),(54,89),(55,90),(56,91),(57,92),(58,93),(59,94),(60,95),(61,96),(62,73),(63,74),(64,75),(65,76),(66,77),(67,78),(68,79),(69,80),(70,81),(71,82),(72,83),(97,167),(98,168),(99,145),(100,146),(101,147),(102,148),(103,149),(104,150),(105,151),(106,152),(107,153),(108,154),(109,155),(110,156),(111,157),(112,158),(113,159),(114,160),(115,161),(116,162),(117,163),(118,164),(119,165),(120,166),(121,260),(122,261),(123,262),(124,263),(125,264),(126,241),(127,242),(128,243),(129,244),(130,245),(131,246),(132,247),(133,248),(134,249),(135,250),(136,251),(137,252),(138,253),(139,254),(140,255),(141,256),(142,257),(143,258),(144,259),(217,281),(218,282),(219,283),(220,284),(221,285),(222,286),(223,287),(224,288),(225,265),(226,266),(227,267),(228,268),(229,269),(230,270),(231,271),(232,272),(233,273),(234,274),(235,275),(236,276),(237,277),(238,278),(239,279),(240,280)], [(1,210,128,272,81,163),(2,211,129,273,82,164),(3,212,130,274,83,165),(4,213,131,275,84,166),(5,214,132,276,85,167),(6,215,133,277,86,168),(7,216,134,278,87,145),(8,193,135,279,88,146),(9,194,136,280,89,147),(10,195,137,281,90,148),(11,196,138,282,91,149),(12,197,139,283,92,150),(13,198,140,284,93,151),(14,199,141,285,94,152),(15,200,142,286,95,153),(16,201,143,287,96,154),(17,202,144,288,73,155),(18,203,121,265,74,156),(19,204,122,266,75,157),(20,205,123,267,76,158),(21,206,124,268,77,159),(22,207,125,269,78,160),(23,208,126,270,79,161),(24,209,127,271,80,162),(25,241,230,68,115,192),(26,242,231,69,116,169),(27,243,232,70,117,170),(28,244,233,71,118,171),(29,245,234,72,119,172),(30,246,235,49,120,173),(31,247,236,50,97,174),(32,248,237,51,98,175),(33,249,238,52,99,176),(34,250,239,53,100,177),(35,251,240,54,101,178),(36,252,217,55,102,179),(37,253,218,56,103,180),(38,254,219,57,104,181),(39,255,220,58,105,182),(40,256,221,59,106,183),(41,257,222,60,107,184),(42,258,223,61,108,185),(43,259,224,62,109,186),(44,260,225,63,110,187),(45,261,226,64,111,188),(46,262,227,65,112,189),(47,263,228,66,113,190),(48,264,229,67,114,191)], [(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···2G3A···3H4A···4H6A···6BD8A···8P12A···12BL24A···24DX
order12···23···34···46···68···812···1224···24
size11···11···11···11···11···11···11···1

288 irreducible representations

dim111111111111
type+++
imageC1C2C2C3C4C4C6C6C8C12C12C24
kernelC2×C6×C24C6×C24C2×C6×C12C22×C24C6×C12C2×C62C2×C24C22×C12C62C2×C12C22×C6C2×C6
# reps161862488164816128

Matrix representation of C2×C6×C24 in GL3(𝔽73) generated by

100
0720
0072
,
6500
010
0065
,
2400
0430
0024
G:=sub<GL(3,GF(73))| [1,0,0,0,72,0,0,0,72],[65,0,0,0,1,0,0,0,65],[24,0,0,0,43,0,0,0,24] >;

C2×C6×C24 in GAP, Magma, Sage, TeX

C_2\times C_6\times C_{24}
% in TeX

G:=Group("C2xC6xC24");
// GroupNames label

G:=SmallGroup(288,826);
// by ID

G=gap.SmallGroup(288,826);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-2,504,124]);
// Polycyclic

G:=Group<a,b,c|a^2=b^6=c^24=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

׿
×
𝔽