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G = C2×C144order 288 = 25·32

Abelian group of type [2,144]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C144, SmallGroup(288,59)

Series: Derived Chief Lower central Upper central

C1 — C2×C144
C1C2C4C12C24C72C144 — C2×C144
C1 — C2×C144
C1 — C2×C144

Generators and relations for C2×C144
 G = < a,b | a2=b144=1, ab=ba >


Smallest permutation representation of C2×C144
Regular action on 288 points
Generators in S288
(1 159)(2 160)(3 161)(4 162)(5 163)(6 164)(7 165)(8 166)(9 167)(10 168)(11 169)(12 170)(13 171)(14 172)(15 173)(16 174)(17 175)(18 176)(19 177)(20 178)(21 179)(22 180)(23 181)(24 182)(25 183)(26 184)(27 185)(28 186)(29 187)(30 188)(31 189)(32 190)(33 191)(34 192)(35 193)(36 194)(37 195)(38 196)(39 197)(40 198)(41 199)(42 200)(43 201)(44 202)(45 203)(46 204)(47 205)(48 206)(49 207)(50 208)(51 209)(52 210)(53 211)(54 212)(55 213)(56 214)(57 215)(58 216)(59 217)(60 218)(61 219)(62 220)(63 221)(64 222)(65 223)(66 224)(67 225)(68 226)(69 227)(70 228)(71 229)(72 230)(73 231)(74 232)(75 233)(76 234)(77 235)(78 236)(79 237)(80 238)(81 239)(82 240)(83 241)(84 242)(85 243)(86 244)(87 245)(88 246)(89 247)(90 248)(91 249)(92 250)(93 251)(94 252)(95 253)(96 254)(97 255)(98 256)(99 257)(100 258)(101 259)(102 260)(103 261)(104 262)(105 263)(106 264)(107 265)(108 266)(109 267)(110 268)(111 269)(112 270)(113 271)(114 272)(115 273)(116 274)(117 275)(118 276)(119 277)(120 278)(121 279)(122 280)(123 281)(124 282)(125 283)(126 284)(127 285)(128 286)(129 287)(130 288)(131 145)(132 146)(133 147)(134 148)(135 149)(136 150)(137 151)(138 152)(139 153)(140 154)(141 155)(142 156)(143 157)(144 158)
(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)| (1,159)(2,160)(3,161)(4,162)(5,163)(6,164)(7,165)(8,166)(9,167)(10,168)(11,169)(12,170)(13,171)(14,172)(15,173)(16,174)(17,175)(18,176)(19,177)(20,178)(21,179)(22,180)(23,181)(24,182)(25,183)(26,184)(27,185)(28,186)(29,187)(30,188)(31,189)(32,190)(33,191)(34,192)(35,193)(36,194)(37,195)(38,196)(39,197)(40,198)(41,199)(42,200)(43,201)(44,202)(45,203)(46,204)(47,205)(48,206)(49,207)(50,208)(51,209)(52,210)(53,211)(54,212)(55,213)(56,214)(57,215)(58,216)(59,217)(60,218)(61,219)(62,220)(63,221)(64,222)(65,223)(66,224)(67,225)(68,226)(69,227)(70,228)(71,229)(72,230)(73,231)(74,232)(75,233)(76,234)(77,235)(78,236)(79,237)(80,238)(81,239)(82,240)(83,241)(84,242)(85,243)(86,244)(87,245)(88,246)(89,247)(90,248)(91,249)(92,250)(93,251)(94,252)(95,253)(96,254)(97,255)(98,256)(99,257)(100,258)(101,259)(102,260)(103,261)(104,262)(105,263)(106,264)(107,265)(108,266)(109,267)(110,268)(111,269)(112,270)(113,271)(114,272)(115,273)(116,274)(117,275)(118,276)(119,277)(120,278)(121,279)(122,280)(123,281)(124,282)(125,283)(126,284)(127,285)(128,286)(129,287)(130,288)(131,145)(132,146)(133,147)(134,148)(135,149)(136,150)(137,151)(138,152)(139,153)(140,154)(141,155)(142,156)(143,157)(144,158), (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( (1,159)(2,160)(3,161)(4,162)(5,163)(6,164)(7,165)(8,166)(9,167)(10,168)(11,169)(12,170)(13,171)(14,172)(15,173)(16,174)(17,175)(18,176)(19,177)(20,178)(21,179)(22,180)(23,181)(24,182)(25,183)(26,184)(27,185)(28,186)(29,187)(30,188)(31,189)(32,190)(33,191)(34,192)(35,193)(36,194)(37,195)(38,196)(39,197)(40,198)(41,199)(42,200)(43,201)(44,202)(45,203)(46,204)(47,205)(48,206)(49,207)(50,208)(51,209)(52,210)(53,211)(54,212)(55,213)(56,214)(57,215)(58,216)(59,217)(60,218)(61,219)(62,220)(63,221)(64,222)(65,223)(66,224)(67,225)(68,226)(69,227)(70,228)(71,229)(72,230)(73,231)(74,232)(75,233)(76,234)(77,235)(78,236)(79,237)(80,238)(81,239)(82,240)(83,241)(84,242)(85,243)(86,244)(87,245)(88,246)(89,247)(90,248)(91,249)(92,250)(93,251)(94,252)(95,253)(96,254)(97,255)(98,256)(99,257)(100,258)(101,259)(102,260)(103,261)(104,262)(105,263)(106,264)(107,265)(108,266)(109,267)(110,268)(111,269)(112,270)(113,271)(114,272)(115,273)(116,274)(117,275)(118,276)(119,277)(120,278)(121,279)(122,280)(123,281)(124,282)(125,283)(126,284)(127,285)(128,286)(129,287)(130,288)(131,145)(132,146)(133,147)(134,148)(135,149)(136,150)(137,151)(138,152)(139,153)(140,154)(141,155)(142,156)(143,157)(144,158), (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([(1,159),(2,160),(3,161),(4,162),(5,163),(6,164),(7,165),(8,166),(9,167),(10,168),(11,169),(12,170),(13,171),(14,172),(15,173),(16,174),(17,175),(18,176),(19,177),(20,178),(21,179),(22,180),(23,181),(24,182),(25,183),(26,184),(27,185),(28,186),(29,187),(30,188),(31,189),(32,190),(33,191),(34,192),(35,193),(36,194),(37,195),(38,196),(39,197),(40,198),(41,199),(42,200),(43,201),(44,202),(45,203),(46,204),(47,205),(48,206),(49,207),(50,208),(51,209),(52,210),(53,211),(54,212),(55,213),(56,214),(57,215),(58,216),(59,217),(60,218),(61,219),(62,220),(63,221),(64,222),(65,223),(66,224),(67,225),(68,226),(69,227),(70,228),(71,229),(72,230),(73,231),(74,232),(75,233),(76,234),(77,235),(78,236),(79,237),(80,238),(81,239),(82,240),(83,241),(84,242),(85,243),(86,244),(87,245),(88,246),(89,247),(90,248),(91,249),(92,250),(93,251),(94,252),(95,253),(96,254),(97,255),(98,256),(99,257),(100,258),(101,259),(102,260),(103,261),(104,262),(105,263),(106,264),(107,265),(108,266),(109,267),(110,268),(111,269),(112,270),(113,271),(114,272),(115,273),(116,274),(117,275),(118,276),(119,277),(120,278),(121,279),(122,280),(123,281),(124,282),(125,283),(126,284),(127,285),(128,286),(129,287),(130,288),(131,145),(132,146),(133,147),(134,148),(135,149),(136,150),(137,151),(138,152),(139,153),(140,154),(141,155),(142,156),(143,157),(144,158)], [(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 2A2B2C3A3B4A4B4C4D6A···6F8A···8H9A···9F12A···12H16A···16P18A···18R24A···24P36A···36X48A···48AF72A···72AV144A···144CR
order12223344446···68···89···912···1216···1618···1824···2436···3648···4872···72144···144
size11111111111···11···11···11···11···11···11···11···11···11···11···1

288 irreducible representations

dim111111111111111111111111
type+++
imageC1C2C2C3C4C4C6C6C8C8C9C12C12C16C18C18C24C24C36C36C48C72C72C144
kernelC2×C144C144C2×C72C2×C48C72C2×C36C48C2×C24C36C2×C18C2×C16C24C2×C12C18C16C2×C8C12C2×C6C8C2×C4C6C4C22C2
# reps12122242446441612688121232242496

Matrix representation of C2×C144 in GL2(𝔽433) generated by

4320
0432
,
2790
02
G:=sub<GL(2,GF(433))| [432,0,0,432],[279,0,0,2] >;

C2×C144 in GAP, Magma, Sage, TeX

C_2\times C_{144}
% in TeX

G:=Group("C2xC144");
// GroupNames label

G:=SmallGroup(288,59);
// by ID

G=gap.SmallGroup(288,59);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-3,-2,-2,84,142,192,124]);
// Polycyclic

G:=Group<a,b|a^2=b^144=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C144 in TeX

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