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G = C22×C74order 296 = 23·37

Abelian group of type [2,2,74]

direct product, abelian, monomial, 2-elementary

Aliases: C22×C74, SmallGroup(296,14)

Series: Derived Chief Lower central Upper central

C1 — C22×C74
C1C37C74C2×C74 — C22×C74
C1 — C22×C74
C1 — C22×C74

Generators and relations for C22×C74
 G = < a,b,c | a2=b2=c74=1, ab=ba, ac=ca, bc=cb >


Smallest permutation representation of C22×C74
Regular action on 296 points
Generators in S296
(1 209)(2 210)(3 211)(4 212)(5 213)(6 214)(7 215)(8 216)(9 217)(10 218)(11 219)(12 220)(13 221)(14 222)(15 149)(16 150)(17 151)(18 152)(19 153)(20 154)(21 155)(22 156)(23 157)(24 158)(25 159)(26 160)(27 161)(28 162)(29 163)(30 164)(31 165)(32 166)(33 167)(34 168)(35 169)(36 170)(37 171)(38 172)(39 173)(40 174)(41 175)(42 176)(43 177)(44 178)(45 179)(46 180)(47 181)(48 182)(49 183)(50 184)(51 185)(52 186)(53 187)(54 188)(55 189)(56 190)(57 191)(58 192)(59 193)(60 194)(61 195)(62 196)(63 197)(64 198)(65 199)(66 200)(67 201)(68 202)(69 203)(70 204)(71 205)(72 206)(73 207)(74 208)(75 246)(76 247)(77 248)(78 249)(79 250)(80 251)(81 252)(82 253)(83 254)(84 255)(85 256)(86 257)(87 258)(88 259)(89 260)(90 261)(91 262)(92 263)(93 264)(94 265)(95 266)(96 267)(97 268)(98 269)(99 270)(100 271)(101 272)(102 273)(103 274)(104 275)(105 276)(106 277)(107 278)(108 279)(109 280)(110 281)(111 282)(112 283)(113 284)(114 285)(115 286)(116 287)(117 288)(118 289)(119 290)(120 291)(121 292)(122 293)(123 294)(124 295)(125 296)(126 223)(127 224)(128 225)(129 226)(130 227)(131 228)(132 229)(133 230)(134 231)(135 232)(136 233)(137 234)(138 235)(139 236)(140 237)(141 238)(142 239)(143 240)(144 241)(145 242)(146 243)(147 244)(148 245)
(1 114)(2 115)(3 116)(4 117)(5 118)(6 119)(7 120)(8 121)(9 122)(10 123)(11 124)(12 125)(13 126)(14 127)(15 128)(16 129)(17 130)(18 131)(19 132)(20 133)(21 134)(22 135)(23 136)(24 137)(25 138)(26 139)(27 140)(28 141)(29 142)(30 143)(31 144)(32 145)(33 146)(34 147)(35 148)(36 75)(37 76)(38 77)(39 78)(40 79)(41 80)(42 81)(43 82)(44 83)(45 84)(46 85)(47 86)(48 87)(49 88)(50 89)(51 90)(52 91)(53 92)(54 93)(55 94)(56 95)(57 96)(58 97)(59 98)(60 99)(61 100)(62 101)(63 102)(64 103)(65 104)(66 105)(67 106)(68 107)(69 108)(70 109)(71 110)(72 111)(73 112)(74 113)(149 225)(150 226)(151 227)(152 228)(153 229)(154 230)(155 231)(156 232)(157 233)(158 234)(159 235)(160 236)(161 237)(162 238)(163 239)(164 240)(165 241)(166 242)(167 243)(168 244)(169 245)(170 246)(171 247)(172 248)(173 249)(174 250)(175 251)(176 252)(177 253)(178 254)(179 255)(180 256)(181 257)(182 258)(183 259)(184 260)(185 261)(186 262)(187 263)(188 264)(189 265)(190 266)(191 267)(192 268)(193 269)(194 270)(195 271)(196 272)(197 273)(198 274)(199 275)(200 276)(201 277)(202 278)(203 279)(204 280)(205 281)(206 282)(207 283)(208 284)(209 285)(210 286)(211 287)(212 288)(213 289)(214 290)(215 291)(216 292)(217 293)(218 294)(219 295)(220 296)(221 223)(222 224)
(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)(223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296)

G:=sub<Sym(296)| (1,209)(2,210)(3,211)(4,212)(5,213)(6,214)(7,215)(8,216)(9,217)(10,218)(11,219)(12,220)(13,221)(14,222)(15,149)(16,150)(17,151)(18,152)(19,153)(20,154)(21,155)(22,156)(23,157)(24,158)(25,159)(26,160)(27,161)(28,162)(29,163)(30,164)(31,165)(32,166)(33,167)(34,168)(35,169)(36,170)(37,171)(38,172)(39,173)(40,174)(41,175)(42,176)(43,177)(44,178)(45,179)(46,180)(47,181)(48,182)(49,183)(50,184)(51,185)(52,186)(53,187)(54,188)(55,189)(56,190)(57,191)(58,192)(59,193)(60,194)(61,195)(62,196)(63,197)(64,198)(65,199)(66,200)(67,201)(68,202)(69,203)(70,204)(71,205)(72,206)(73,207)(74,208)(75,246)(76,247)(77,248)(78,249)(79,250)(80,251)(81,252)(82,253)(83,254)(84,255)(85,256)(86,257)(87,258)(88,259)(89,260)(90,261)(91,262)(92,263)(93,264)(94,265)(95,266)(96,267)(97,268)(98,269)(99,270)(100,271)(101,272)(102,273)(103,274)(104,275)(105,276)(106,277)(107,278)(108,279)(109,280)(110,281)(111,282)(112,283)(113,284)(114,285)(115,286)(116,287)(117,288)(118,289)(119,290)(120,291)(121,292)(122,293)(123,294)(124,295)(125,296)(126,223)(127,224)(128,225)(129,226)(130,227)(131,228)(132,229)(133,230)(134,231)(135,232)(136,233)(137,234)(138,235)(139,236)(140,237)(141,238)(142,239)(143,240)(144,241)(145,242)(146,243)(147,244)(148,245), (1,114)(2,115)(3,116)(4,117)(5,118)(6,119)(7,120)(8,121)(9,122)(10,123)(11,124)(12,125)(13,126)(14,127)(15,128)(16,129)(17,130)(18,131)(19,132)(20,133)(21,134)(22,135)(23,136)(24,137)(25,138)(26,139)(27,140)(28,141)(29,142)(30,143)(31,144)(32,145)(33,146)(34,147)(35,148)(36,75)(37,76)(38,77)(39,78)(40,79)(41,80)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,88)(50,89)(51,90)(52,91)(53,92)(54,93)(55,94)(56,95)(57,96)(58,97)(59,98)(60,99)(61,100)(62,101)(63,102)(64,103)(65,104)(66,105)(67,106)(68,107)(69,108)(70,109)(71,110)(72,111)(73,112)(74,113)(149,225)(150,226)(151,227)(152,228)(153,229)(154,230)(155,231)(156,232)(157,233)(158,234)(159,235)(160,236)(161,237)(162,238)(163,239)(164,240)(165,241)(166,242)(167,243)(168,244)(169,245)(170,246)(171,247)(172,248)(173,249)(174,250)(175,251)(176,252)(177,253)(178,254)(179,255)(180,256)(181,257)(182,258)(183,259)(184,260)(185,261)(186,262)(187,263)(188,264)(189,265)(190,266)(191,267)(192,268)(193,269)(194,270)(195,271)(196,272)(197,273)(198,274)(199,275)(200,276)(201,277)(202,278)(203,279)(204,280)(205,281)(206,282)(207,283)(208,284)(209,285)(210,286)(211,287)(212,288)(213,289)(214,290)(215,291)(216,292)(217,293)(218,294)(219,295)(220,296)(221,223)(222,224), (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)(223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296)>;

G:=Group( (1,209)(2,210)(3,211)(4,212)(5,213)(6,214)(7,215)(8,216)(9,217)(10,218)(11,219)(12,220)(13,221)(14,222)(15,149)(16,150)(17,151)(18,152)(19,153)(20,154)(21,155)(22,156)(23,157)(24,158)(25,159)(26,160)(27,161)(28,162)(29,163)(30,164)(31,165)(32,166)(33,167)(34,168)(35,169)(36,170)(37,171)(38,172)(39,173)(40,174)(41,175)(42,176)(43,177)(44,178)(45,179)(46,180)(47,181)(48,182)(49,183)(50,184)(51,185)(52,186)(53,187)(54,188)(55,189)(56,190)(57,191)(58,192)(59,193)(60,194)(61,195)(62,196)(63,197)(64,198)(65,199)(66,200)(67,201)(68,202)(69,203)(70,204)(71,205)(72,206)(73,207)(74,208)(75,246)(76,247)(77,248)(78,249)(79,250)(80,251)(81,252)(82,253)(83,254)(84,255)(85,256)(86,257)(87,258)(88,259)(89,260)(90,261)(91,262)(92,263)(93,264)(94,265)(95,266)(96,267)(97,268)(98,269)(99,270)(100,271)(101,272)(102,273)(103,274)(104,275)(105,276)(106,277)(107,278)(108,279)(109,280)(110,281)(111,282)(112,283)(113,284)(114,285)(115,286)(116,287)(117,288)(118,289)(119,290)(120,291)(121,292)(122,293)(123,294)(124,295)(125,296)(126,223)(127,224)(128,225)(129,226)(130,227)(131,228)(132,229)(133,230)(134,231)(135,232)(136,233)(137,234)(138,235)(139,236)(140,237)(141,238)(142,239)(143,240)(144,241)(145,242)(146,243)(147,244)(148,245), (1,114)(2,115)(3,116)(4,117)(5,118)(6,119)(7,120)(8,121)(9,122)(10,123)(11,124)(12,125)(13,126)(14,127)(15,128)(16,129)(17,130)(18,131)(19,132)(20,133)(21,134)(22,135)(23,136)(24,137)(25,138)(26,139)(27,140)(28,141)(29,142)(30,143)(31,144)(32,145)(33,146)(34,147)(35,148)(36,75)(37,76)(38,77)(39,78)(40,79)(41,80)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,88)(50,89)(51,90)(52,91)(53,92)(54,93)(55,94)(56,95)(57,96)(58,97)(59,98)(60,99)(61,100)(62,101)(63,102)(64,103)(65,104)(66,105)(67,106)(68,107)(69,108)(70,109)(71,110)(72,111)(73,112)(74,113)(149,225)(150,226)(151,227)(152,228)(153,229)(154,230)(155,231)(156,232)(157,233)(158,234)(159,235)(160,236)(161,237)(162,238)(163,239)(164,240)(165,241)(166,242)(167,243)(168,244)(169,245)(170,246)(171,247)(172,248)(173,249)(174,250)(175,251)(176,252)(177,253)(178,254)(179,255)(180,256)(181,257)(182,258)(183,259)(184,260)(185,261)(186,262)(187,263)(188,264)(189,265)(190,266)(191,267)(192,268)(193,269)(194,270)(195,271)(196,272)(197,273)(198,274)(199,275)(200,276)(201,277)(202,278)(203,279)(204,280)(205,281)(206,282)(207,283)(208,284)(209,285)(210,286)(211,287)(212,288)(213,289)(214,290)(215,291)(216,292)(217,293)(218,294)(219,295)(220,296)(221,223)(222,224), (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)(223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296) );

G=PermutationGroup([(1,209),(2,210),(3,211),(4,212),(5,213),(6,214),(7,215),(8,216),(9,217),(10,218),(11,219),(12,220),(13,221),(14,222),(15,149),(16,150),(17,151),(18,152),(19,153),(20,154),(21,155),(22,156),(23,157),(24,158),(25,159),(26,160),(27,161),(28,162),(29,163),(30,164),(31,165),(32,166),(33,167),(34,168),(35,169),(36,170),(37,171),(38,172),(39,173),(40,174),(41,175),(42,176),(43,177),(44,178),(45,179),(46,180),(47,181),(48,182),(49,183),(50,184),(51,185),(52,186),(53,187),(54,188),(55,189),(56,190),(57,191),(58,192),(59,193),(60,194),(61,195),(62,196),(63,197),(64,198),(65,199),(66,200),(67,201),(68,202),(69,203),(70,204),(71,205),(72,206),(73,207),(74,208),(75,246),(76,247),(77,248),(78,249),(79,250),(80,251),(81,252),(82,253),(83,254),(84,255),(85,256),(86,257),(87,258),(88,259),(89,260),(90,261),(91,262),(92,263),(93,264),(94,265),(95,266),(96,267),(97,268),(98,269),(99,270),(100,271),(101,272),(102,273),(103,274),(104,275),(105,276),(106,277),(107,278),(108,279),(109,280),(110,281),(111,282),(112,283),(113,284),(114,285),(115,286),(116,287),(117,288),(118,289),(119,290),(120,291),(121,292),(122,293),(123,294),(124,295),(125,296),(126,223),(127,224),(128,225),(129,226),(130,227),(131,228),(132,229),(133,230),(134,231),(135,232),(136,233),(137,234),(138,235),(139,236),(140,237),(141,238),(142,239),(143,240),(144,241),(145,242),(146,243),(147,244),(148,245)], [(1,114),(2,115),(3,116),(4,117),(5,118),(6,119),(7,120),(8,121),(9,122),(10,123),(11,124),(12,125),(13,126),(14,127),(15,128),(16,129),(17,130),(18,131),(19,132),(20,133),(21,134),(22,135),(23,136),(24,137),(25,138),(26,139),(27,140),(28,141),(29,142),(30,143),(31,144),(32,145),(33,146),(34,147),(35,148),(36,75),(37,76),(38,77),(39,78),(40,79),(41,80),(42,81),(43,82),(44,83),(45,84),(46,85),(47,86),(48,87),(49,88),(50,89),(51,90),(52,91),(53,92),(54,93),(55,94),(56,95),(57,96),(58,97),(59,98),(60,99),(61,100),(62,101),(63,102),(64,103),(65,104),(66,105),(67,106),(68,107),(69,108),(70,109),(71,110),(72,111),(73,112),(74,113),(149,225),(150,226),(151,227),(152,228),(153,229),(154,230),(155,231),(156,232),(157,233),(158,234),(159,235),(160,236),(161,237),(162,238),(163,239),(164,240),(165,241),(166,242),(167,243),(168,244),(169,245),(170,246),(171,247),(172,248),(173,249),(174,250),(175,251),(176,252),(177,253),(178,254),(179,255),(180,256),(181,257),(182,258),(183,259),(184,260),(185,261),(186,262),(187,263),(188,264),(189,265),(190,266),(191,267),(192,268),(193,269),(194,270),(195,271),(196,272),(197,273),(198,274),(199,275),(200,276),(201,277),(202,278),(203,279),(204,280),(205,281),(206,282),(207,283),(208,284),(209,285),(210,286),(211,287),(212,288),(213,289),(214,290),(215,291),(216,292),(217,293),(218,294),(219,295),(220,296),(221,223),(222,224)], [(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),(223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296)])

296 conjugacy classes

class 1 2A···2G37A···37AJ74A···74IR
order12···237···3774···74
size11···11···11···1

296 irreducible representations

dim1111
type++
imageC1C2C37C74
kernelC22×C74C2×C74C23C22
# reps1736252

Matrix representation of C22×C74 in GL3(𝔽149) generated by

100
010
00148
,
100
01480
001
,
11000
01230
00142
G:=sub<GL(3,GF(149))| [1,0,0,0,1,0,0,0,148],[1,0,0,0,148,0,0,0,1],[110,0,0,0,123,0,0,0,142] >;

C22×C74 in GAP, Magma, Sage, TeX

C_2^2\times C_{74}
% in TeX

G:=Group("C2^2xC74");
// GroupNames label

G:=SmallGroup(296,14);
// by ID

G=gap.SmallGroup(296,14);
# by ID

G:=PCGroup([4,-2,-2,-2,-37]);
// Polycyclic

G:=Group<a,b,c|a^2=b^2=c^74=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

Export

Subgroup lattice of C22×C74 in TeX

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