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G = C22×C72order 288 = 25·32

Abelian group of type [2,2,72]

direct product, abelian, monomial, 2-elementary

Aliases: C22×C72, SmallGroup(288,179)

Series: Derived Chief Lower central Upper central

C1 — C22×C72
C1C2C6C12C36C72C2×C72 — C22×C72
C1 — C22×C72
C1 — C22×C72

Generators and relations for C22×C72
 G = < a,b,c | a2=b2=c72=1, ab=ba, ac=ca, bc=cb >

Subgroups: 114, all normal (18 characteristic)
C1, C2, C2 [×6], C3, C4, C4 [×3], C22 [×7], C6, C6 [×6], C8 [×4], C2×C4 [×6], C23, C9, C12, C12 [×3], C2×C6 [×7], C2×C8 [×6], C22×C4, C18, C18 [×6], C24 [×4], C2×C12 [×6], C22×C6, C22×C8, C36, C36 [×3], C2×C18 [×7], C2×C24 [×6], C22×C12, C72 [×4], C2×C36 [×6], C22×C18, C22×C24, C2×C72 [×6], C22×C36, C22×C72
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C8 [×4], C2×C4 [×6], C23, C9, C12 [×4], C2×C6 [×7], C2×C8 [×6], C22×C4, C18 [×7], C24 [×4], C2×C12 [×6], C22×C6, C22×C8, C36 [×4], C2×C18 [×7], C2×C24 [×6], C22×C12, C72 [×4], C2×C36 [×6], C22×C18, C22×C24, C2×C72 [×6], C22×C36, C22×C72

Smallest permutation representation of C22×C72
Regular action on 288 points
Generators in S288
(1 221)(2 222)(3 223)(4 224)(5 225)(6 226)(7 227)(8 228)(9 229)(10 230)(11 231)(12 232)(13 233)(14 234)(15 235)(16 236)(17 237)(18 238)(19 239)(20 240)(21 241)(22 242)(23 243)(24 244)(25 245)(26 246)(27 247)(28 248)(29 249)(30 250)(31 251)(32 252)(33 253)(34 254)(35 255)(36 256)(37 257)(38 258)(39 259)(40 260)(41 261)(42 262)(43 263)(44 264)(45 265)(46 266)(47 267)(48 268)(49 269)(50 270)(51 271)(52 272)(53 273)(54 274)(55 275)(56 276)(57 277)(58 278)(59 279)(60 280)(61 281)(62 282)(63 283)(64 284)(65 285)(66 286)(67 287)(68 288)(69 217)(70 218)(71 219)(72 220)(73 199)(74 200)(75 201)(76 202)(77 203)(78 204)(79 205)(80 206)(81 207)(82 208)(83 209)(84 210)(85 211)(86 212)(87 213)(88 214)(89 215)(90 216)(91 145)(92 146)(93 147)(94 148)(95 149)(96 150)(97 151)(98 152)(99 153)(100 154)(101 155)(102 156)(103 157)(104 158)(105 159)(106 160)(107 161)(108 162)(109 163)(110 164)(111 165)(112 166)(113 167)(114 168)(115 169)(116 170)(117 171)(118 172)(119 173)(120 174)(121 175)(122 176)(123 177)(124 178)(125 179)(126 180)(127 181)(128 182)(129 183)(130 184)(131 185)(132 186)(133 187)(134 188)(135 189)(136 190)(137 191)(138 192)(139 193)(140 194)(141 195)(142 196)(143 197)(144 198)
(1 156)(2 157)(3 158)(4 159)(5 160)(6 161)(7 162)(8 163)(9 164)(10 165)(11 166)(12 167)(13 168)(14 169)(15 170)(16 171)(17 172)(18 173)(19 174)(20 175)(21 176)(22 177)(23 178)(24 179)(25 180)(26 181)(27 182)(28 183)(29 184)(30 185)(31 186)(32 187)(33 188)(34 189)(35 190)(36 191)(37 192)(38 193)(39 194)(40 195)(41 196)(42 197)(43 198)(44 199)(45 200)(46 201)(47 202)(48 203)(49 204)(50 205)(51 206)(52 207)(53 208)(54 209)(55 210)(56 211)(57 212)(58 213)(59 214)(60 215)(61 216)(62 145)(63 146)(64 147)(65 148)(66 149)(67 150)(68 151)(69 152)(70 153)(71 154)(72 155)(73 264)(74 265)(75 266)(76 267)(77 268)(78 269)(79 270)(80 271)(81 272)(82 273)(83 274)(84 275)(85 276)(86 277)(87 278)(88 279)(89 280)(90 281)(91 282)(92 283)(93 284)(94 285)(95 286)(96 287)(97 288)(98 217)(99 218)(100 219)(101 220)(102 221)(103 222)(104 223)(105 224)(106 225)(107 226)(108 227)(109 228)(110 229)(111 230)(112 231)(113 232)(114 233)(115 234)(116 235)(117 236)(118 237)(119 238)(120 239)(121 240)(122 241)(123 242)(124 243)(125 244)(126 245)(127 246)(128 247)(129 248)(130 249)(131 250)(132 251)(133 252)(134 253)(135 254)(136 255)(137 256)(138 257)(139 258)(140 259)(141 260)(142 261)(143 262)(144 263)
(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)| (1,221)(2,222)(3,223)(4,224)(5,225)(6,226)(7,227)(8,228)(9,229)(10,230)(11,231)(12,232)(13,233)(14,234)(15,235)(16,236)(17,237)(18,238)(19,239)(20,240)(21,241)(22,242)(23,243)(24,244)(25,245)(26,246)(27,247)(28,248)(29,249)(30,250)(31,251)(32,252)(33,253)(34,254)(35,255)(36,256)(37,257)(38,258)(39,259)(40,260)(41,261)(42,262)(43,263)(44,264)(45,265)(46,266)(47,267)(48,268)(49,269)(50,270)(51,271)(52,272)(53,273)(54,274)(55,275)(56,276)(57,277)(58,278)(59,279)(60,280)(61,281)(62,282)(63,283)(64,284)(65,285)(66,286)(67,287)(68,288)(69,217)(70,218)(71,219)(72,220)(73,199)(74,200)(75,201)(76,202)(77,203)(78,204)(79,205)(80,206)(81,207)(82,208)(83,209)(84,210)(85,211)(86,212)(87,213)(88,214)(89,215)(90,216)(91,145)(92,146)(93,147)(94,148)(95,149)(96,150)(97,151)(98,152)(99,153)(100,154)(101,155)(102,156)(103,157)(104,158)(105,159)(106,160)(107,161)(108,162)(109,163)(110,164)(111,165)(112,166)(113,167)(114,168)(115,169)(116,170)(117,171)(118,172)(119,173)(120,174)(121,175)(122,176)(123,177)(124,178)(125,179)(126,180)(127,181)(128,182)(129,183)(130,184)(131,185)(132,186)(133,187)(134,188)(135,189)(136,190)(137,191)(138,192)(139,193)(140,194)(141,195)(142,196)(143,197)(144,198), (1,156)(2,157)(3,158)(4,159)(5,160)(6,161)(7,162)(8,163)(9,164)(10,165)(11,166)(12,167)(13,168)(14,169)(15,170)(16,171)(17,172)(18,173)(19,174)(20,175)(21,176)(22,177)(23,178)(24,179)(25,180)(26,181)(27,182)(28,183)(29,184)(30,185)(31,186)(32,187)(33,188)(34,189)(35,190)(36,191)(37,192)(38,193)(39,194)(40,195)(41,196)(42,197)(43,198)(44,199)(45,200)(46,201)(47,202)(48,203)(49,204)(50,205)(51,206)(52,207)(53,208)(54,209)(55,210)(56,211)(57,212)(58,213)(59,214)(60,215)(61,216)(62,145)(63,146)(64,147)(65,148)(66,149)(67,150)(68,151)(69,152)(70,153)(71,154)(72,155)(73,264)(74,265)(75,266)(76,267)(77,268)(78,269)(79,270)(80,271)(81,272)(82,273)(83,274)(84,275)(85,276)(86,277)(87,278)(88,279)(89,280)(90,281)(91,282)(92,283)(93,284)(94,285)(95,286)(96,287)(97,288)(98,217)(99,218)(100,219)(101,220)(102,221)(103,222)(104,223)(105,224)(106,225)(107,226)(108,227)(109,228)(110,229)(111,230)(112,231)(113,232)(114,233)(115,234)(116,235)(117,236)(118,237)(119,238)(120,239)(121,240)(122,241)(123,242)(124,243)(125,244)(126,245)(127,246)(128,247)(129,248)(130,249)(131,250)(132,251)(133,252)(134,253)(135,254)(136,255)(137,256)(138,257)(139,258)(140,259)(141,260)(142,261)(143,262)(144,263), (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( (1,221)(2,222)(3,223)(4,224)(5,225)(6,226)(7,227)(8,228)(9,229)(10,230)(11,231)(12,232)(13,233)(14,234)(15,235)(16,236)(17,237)(18,238)(19,239)(20,240)(21,241)(22,242)(23,243)(24,244)(25,245)(26,246)(27,247)(28,248)(29,249)(30,250)(31,251)(32,252)(33,253)(34,254)(35,255)(36,256)(37,257)(38,258)(39,259)(40,260)(41,261)(42,262)(43,263)(44,264)(45,265)(46,266)(47,267)(48,268)(49,269)(50,270)(51,271)(52,272)(53,273)(54,274)(55,275)(56,276)(57,277)(58,278)(59,279)(60,280)(61,281)(62,282)(63,283)(64,284)(65,285)(66,286)(67,287)(68,288)(69,217)(70,218)(71,219)(72,220)(73,199)(74,200)(75,201)(76,202)(77,203)(78,204)(79,205)(80,206)(81,207)(82,208)(83,209)(84,210)(85,211)(86,212)(87,213)(88,214)(89,215)(90,216)(91,145)(92,146)(93,147)(94,148)(95,149)(96,150)(97,151)(98,152)(99,153)(100,154)(101,155)(102,156)(103,157)(104,158)(105,159)(106,160)(107,161)(108,162)(109,163)(110,164)(111,165)(112,166)(113,167)(114,168)(115,169)(116,170)(117,171)(118,172)(119,173)(120,174)(121,175)(122,176)(123,177)(124,178)(125,179)(126,180)(127,181)(128,182)(129,183)(130,184)(131,185)(132,186)(133,187)(134,188)(135,189)(136,190)(137,191)(138,192)(139,193)(140,194)(141,195)(142,196)(143,197)(144,198), (1,156)(2,157)(3,158)(4,159)(5,160)(6,161)(7,162)(8,163)(9,164)(10,165)(11,166)(12,167)(13,168)(14,169)(15,170)(16,171)(17,172)(18,173)(19,174)(20,175)(21,176)(22,177)(23,178)(24,179)(25,180)(26,181)(27,182)(28,183)(29,184)(30,185)(31,186)(32,187)(33,188)(34,189)(35,190)(36,191)(37,192)(38,193)(39,194)(40,195)(41,196)(42,197)(43,198)(44,199)(45,200)(46,201)(47,202)(48,203)(49,204)(50,205)(51,206)(52,207)(53,208)(54,209)(55,210)(56,211)(57,212)(58,213)(59,214)(60,215)(61,216)(62,145)(63,146)(64,147)(65,148)(66,149)(67,150)(68,151)(69,152)(70,153)(71,154)(72,155)(73,264)(74,265)(75,266)(76,267)(77,268)(78,269)(79,270)(80,271)(81,272)(82,273)(83,274)(84,275)(85,276)(86,277)(87,278)(88,279)(89,280)(90,281)(91,282)(92,283)(93,284)(94,285)(95,286)(96,287)(97,288)(98,217)(99,218)(100,219)(101,220)(102,221)(103,222)(104,223)(105,224)(106,225)(107,226)(108,227)(109,228)(110,229)(111,230)(112,231)(113,232)(114,233)(115,234)(116,235)(117,236)(118,237)(119,238)(120,239)(121,240)(122,241)(123,242)(124,243)(125,244)(126,245)(127,246)(128,247)(129,248)(130,249)(131,250)(132,251)(133,252)(134,253)(135,254)(136,255)(137,256)(138,257)(139,258)(140,259)(141,260)(142,261)(143,262)(144,263), (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([(1,221),(2,222),(3,223),(4,224),(5,225),(6,226),(7,227),(8,228),(9,229),(10,230),(11,231),(12,232),(13,233),(14,234),(15,235),(16,236),(17,237),(18,238),(19,239),(20,240),(21,241),(22,242),(23,243),(24,244),(25,245),(26,246),(27,247),(28,248),(29,249),(30,250),(31,251),(32,252),(33,253),(34,254),(35,255),(36,256),(37,257),(38,258),(39,259),(40,260),(41,261),(42,262),(43,263),(44,264),(45,265),(46,266),(47,267),(48,268),(49,269),(50,270),(51,271),(52,272),(53,273),(54,274),(55,275),(56,276),(57,277),(58,278),(59,279),(60,280),(61,281),(62,282),(63,283),(64,284),(65,285),(66,286),(67,287),(68,288),(69,217),(70,218),(71,219),(72,220),(73,199),(74,200),(75,201),(76,202),(77,203),(78,204),(79,205),(80,206),(81,207),(82,208),(83,209),(84,210),(85,211),(86,212),(87,213),(88,214),(89,215),(90,216),(91,145),(92,146),(93,147),(94,148),(95,149),(96,150),(97,151),(98,152),(99,153),(100,154),(101,155),(102,156),(103,157),(104,158),(105,159),(106,160),(107,161),(108,162),(109,163),(110,164),(111,165),(112,166),(113,167),(114,168),(115,169),(116,170),(117,171),(118,172),(119,173),(120,174),(121,175),(122,176),(123,177),(124,178),(125,179),(126,180),(127,181),(128,182),(129,183),(130,184),(131,185),(132,186),(133,187),(134,188),(135,189),(136,190),(137,191),(138,192),(139,193),(140,194),(141,195),(142,196),(143,197),(144,198)], [(1,156),(2,157),(3,158),(4,159),(5,160),(6,161),(7,162),(8,163),(9,164),(10,165),(11,166),(12,167),(13,168),(14,169),(15,170),(16,171),(17,172),(18,173),(19,174),(20,175),(21,176),(22,177),(23,178),(24,179),(25,180),(26,181),(27,182),(28,183),(29,184),(30,185),(31,186),(32,187),(33,188),(34,189),(35,190),(36,191),(37,192),(38,193),(39,194),(40,195),(41,196),(42,197),(43,198),(44,199),(45,200),(46,201),(47,202),(48,203),(49,204),(50,205),(51,206),(52,207),(53,208),(54,209),(55,210),(56,211),(57,212),(58,213),(59,214),(60,215),(61,216),(62,145),(63,146),(64,147),(65,148),(66,149),(67,150),(68,151),(69,152),(70,153),(71,154),(72,155),(73,264),(74,265),(75,266),(76,267),(77,268),(78,269),(79,270),(80,271),(81,272),(82,273),(83,274),(84,275),(85,276),(86,277),(87,278),(88,279),(89,280),(90,281),(91,282),(92,283),(93,284),(94,285),(95,286),(96,287),(97,288),(98,217),(99,218),(100,219),(101,220),(102,221),(103,222),(104,223),(105,224),(106,225),(107,226),(108,227),(109,228),(110,229),(111,230),(112,231),(113,232),(114,233),(115,234),(116,235),(117,236),(118,237),(119,238),(120,239),(121,240),(122,241),(123,242),(124,243),(125,244),(126,245),(127,246),(128,247),(129,248),(130,249),(131,250),(132,251),(133,252),(134,253),(135,254),(136,255),(137,256),(138,257),(139,258),(140,259),(141,260),(142,261),(143,262),(144,263)], [(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···2G3A3B4A···4H6A···6N8A···8P9A···9F12A···12P18A···18AP24A···24AF36A···36AV72A···72CR
order12···2334···46···68···89···912···1218···1824···2436···3672···72
size11···1111···11···11···11···11···11···11···11···11···1

288 irreducible representations

dim111111111111111111
type+++
imageC1C2C2C3C4C4C6C6C8C9C12C12C18C18C24C36C36C72
kernelC22×C72C2×C72C22×C36C22×C24C2×C36C22×C18C2×C24C22×C12C2×C18C22×C8C2×C12C22×C6C2×C8C22×C4C2×C6C2×C4C23C22
# reps16126212216612436632361296

Matrix representation of C22×C72 in GL3(𝔽73) generated by

100
0720
0072
,
100
0720
001
,
6000
0430
0061
G:=sub<GL(3,GF(73))| [1,0,0,0,72,0,0,0,72],[1,0,0,0,72,0,0,0,1],[60,0,0,0,43,0,0,0,61] >;

C22×C72 in GAP, Magma, Sage, TeX

C_2^2\times C_{72}
% in TeX

G:=Group("C2^2xC72");
// GroupNames label

G:=SmallGroup(288,179);
// by ID

G=gap.SmallGroup(288,179);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-2,168,192,242]);
// Polycyclic

G:=Group<a,b,c|a^2=b^2=c^72=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

׿
×
𝔽