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G = C22×C76order 304 = 24·19

Abelian group of type [2,2,76]

direct product, abelian, monomial, 2-elementary

Aliases: C22×C76, SmallGroup(304,37)

Series: Derived Chief Lower central Upper central

C1 — C22×C76
C1C2C38C76C2×C76 — C22×C76
C1 — C22×C76
C1 — C22×C76

Generators and relations for C22×C76
 G = < a,b,c | a2=b2=c76=1, ab=ba, ac=ca, bc=cb >

Subgroups: 54, all normal (8 characteristic)
C1, C2, C2 [×6], C4 [×4], C22 [×7], C2×C4 [×6], C23, C22×C4, C19, C38, C38 [×6], C76 [×4], C2×C38 [×7], C2×C76 [×6], C22×C38, C22×C76
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, C22×C4, C19, C38 [×7], C76 [×4], C2×C38 [×7], C2×C76 [×6], C22×C38, C22×C76

Smallest permutation representation of C22×C76
Regular action on 304 points
Generators in S304
(1 245)(2 246)(3 247)(4 248)(5 249)(6 250)(7 251)(8 252)(9 253)(10 254)(11 255)(12 256)(13 257)(14 258)(15 259)(16 260)(17 261)(18 262)(19 263)(20 264)(21 265)(22 266)(23 267)(24 268)(25 269)(26 270)(27 271)(28 272)(29 273)(30 274)(31 275)(32 276)(33 277)(34 278)(35 279)(36 280)(37 281)(38 282)(39 283)(40 284)(41 285)(42 286)(43 287)(44 288)(45 289)(46 290)(47 291)(48 292)(49 293)(50 294)(51 295)(52 296)(53 297)(54 298)(55 299)(56 300)(57 301)(58 302)(59 303)(60 304)(61 229)(62 230)(63 231)(64 232)(65 233)(66 234)(67 235)(68 236)(69 237)(70 238)(71 239)(72 240)(73 241)(74 242)(75 243)(76 244)(77 182)(78 183)(79 184)(80 185)(81 186)(82 187)(83 188)(84 189)(85 190)(86 191)(87 192)(88 193)(89 194)(90 195)(91 196)(92 197)(93 198)(94 199)(95 200)(96 201)(97 202)(98 203)(99 204)(100 205)(101 206)(102 207)(103 208)(104 209)(105 210)(106 211)(107 212)(108 213)(109 214)(110 215)(111 216)(112 217)(113 218)(114 219)(115 220)(116 221)(117 222)(118 223)(119 224)(120 225)(121 226)(122 227)(123 228)(124 153)(125 154)(126 155)(127 156)(128 157)(129 158)(130 159)(131 160)(132 161)(133 162)(134 163)(135 164)(136 165)(137 166)(138 167)(139 168)(140 169)(141 170)(142 171)(143 172)(144 173)(145 174)(146 175)(147 176)(148 177)(149 178)(150 179)(151 180)(152 181)
(1 183)(2 184)(3 185)(4 186)(5 187)(6 188)(7 189)(8 190)(9 191)(10 192)(11 193)(12 194)(13 195)(14 196)(15 197)(16 198)(17 199)(18 200)(19 201)(20 202)(21 203)(22 204)(23 205)(24 206)(25 207)(26 208)(27 209)(28 210)(29 211)(30 212)(31 213)(32 214)(33 215)(34 216)(35 217)(36 218)(37 219)(38 220)(39 221)(40 222)(41 223)(42 224)(43 225)(44 226)(45 227)(46 228)(47 153)(48 154)(49 155)(50 156)(51 157)(52 158)(53 159)(54 160)(55 161)(56 162)(57 163)(58 164)(59 165)(60 166)(61 167)(62 168)(63 169)(64 170)(65 171)(66 172)(67 173)(68 174)(69 175)(70 176)(71 177)(72 178)(73 179)(74 180)(75 181)(76 182)(77 244)(78 245)(79 246)(80 247)(81 248)(82 249)(83 250)(84 251)(85 252)(86 253)(87 254)(88 255)(89 256)(90 257)(91 258)(92 259)(93 260)(94 261)(95 262)(96 263)(97 264)(98 265)(99 266)(100 267)(101 268)(102 269)(103 270)(104 271)(105 272)(106 273)(107 274)(108 275)(109 276)(110 277)(111 278)(112 279)(113 280)(114 281)(115 282)(116 283)(117 284)(118 285)(119 286)(120 287)(121 288)(122 289)(123 290)(124 291)(125 292)(126 293)(127 294)(128 295)(129 296)(130 297)(131 298)(132 299)(133 300)(134 301)(135 302)(136 303)(137 304)(138 229)(139 230)(140 231)(141 232)(142 233)(143 234)(144 235)(145 236)(146 237)(147 238)(148 239)(149 240)(150 241)(151 242)(152 243)
(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)(229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304)

G:=sub<Sym(304)| (1,245)(2,246)(3,247)(4,248)(5,249)(6,250)(7,251)(8,252)(9,253)(10,254)(11,255)(12,256)(13,257)(14,258)(15,259)(16,260)(17,261)(18,262)(19,263)(20,264)(21,265)(22,266)(23,267)(24,268)(25,269)(26,270)(27,271)(28,272)(29,273)(30,274)(31,275)(32,276)(33,277)(34,278)(35,279)(36,280)(37,281)(38,282)(39,283)(40,284)(41,285)(42,286)(43,287)(44,288)(45,289)(46,290)(47,291)(48,292)(49,293)(50,294)(51,295)(52,296)(53,297)(54,298)(55,299)(56,300)(57,301)(58,302)(59,303)(60,304)(61,229)(62,230)(63,231)(64,232)(65,233)(66,234)(67,235)(68,236)(69,237)(70,238)(71,239)(72,240)(73,241)(74,242)(75,243)(76,244)(77,182)(78,183)(79,184)(80,185)(81,186)(82,187)(83,188)(84,189)(85,190)(86,191)(87,192)(88,193)(89,194)(90,195)(91,196)(92,197)(93,198)(94,199)(95,200)(96,201)(97,202)(98,203)(99,204)(100,205)(101,206)(102,207)(103,208)(104,209)(105,210)(106,211)(107,212)(108,213)(109,214)(110,215)(111,216)(112,217)(113,218)(114,219)(115,220)(116,221)(117,222)(118,223)(119,224)(120,225)(121,226)(122,227)(123,228)(124,153)(125,154)(126,155)(127,156)(128,157)(129,158)(130,159)(131,160)(132,161)(133,162)(134,163)(135,164)(136,165)(137,166)(138,167)(139,168)(140,169)(141,170)(142,171)(143,172)(144,173)(145,174)(146,175)(147,176)(148,177)(149,178)(150,179)(151,180)(152,181), (1,183)(2,184)(3,185)(4,186)(5,187)(6,188)(7,189)(8,190)(9,191)(10,192)(11,193)(12,194)(13,195)(14,196)(15,197)(16,198)(17,199)(18,200)(19,201)(20,202)(21,203)(22,204)(23,205)(24,206)(25,207)(26,208)(27,209)(28,210)(29,211)(30,212)(31,213)(32,214)(33,215)(34,216)(35,217)(36,218)(37,219)(38,220)(39,221)(40,222)(41,223)(42,224)(43,225)(44,226)(45,227)(46,228)(47,153)(48,154)(49,155)(50,156)(51,157)(52,158)(53,159)(54,160)(55,161)(56,162)(57,163)(58,164)(59,165)(60,166)(61,167)(62,168)(63,169)(64,170)(65,171)(66,172)(67,173)(68,174)(69,175)(70,176)(71,177)(72,178)(73,179)(74,180)(75,181)(76,182)(77,244)(78,245)(79,246)(80,247)(81,248)(82,249)(83,250)(84,251)(85,252)(86,253)(87,254)(88,255)(89,256)(90,257)(91,258)(92,259)(93,260)(94,261)(95,262)(96,263)(97,264)(98,265)(99,266)(100,267)(101,268)(102,269)(103,270)(104,271)(105,272)(106,273)(107,274)(108,275)(109,276)(110,277)(111,278)(112,279)(113,280)(114,281)(115,282)(116,283)(117,284)(118,285)(119,286)(120,287)(121,288)(122,289)(123,290)(124,291)(125,292)(126,293)(127,294)(128,295)(129,296)(130,297)(131,298)(132,299)(133,300)(134,301)(135,302)(136,303)(137,304)(138,229)(139,230)(140,231)(141,232)(142,233)(143,234)(144,235)(145,236)(146,237)(147,238)(148,239)(149,240)(150,241)(151,242)(152,243), (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)(229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304)>;

G:=Group( (1,245)(2,246)(3,247)(4,248)(5,249)(6,250)(7,251)(8,252)(9,253)(10,254)(11,255)(12,256)(13,257)(14,258)(15,259)(16,260)(17,261)(18,262)(19,263)(20,264)(21,265)(22,266)(23,267)(24,268)(25,269)(26,270)(27,271)(28,272)(29,273)(30,274)(31,275)(32,276)(33,277)(34,278)(35,279)(36,280)(37,281)(38,282)(39,283)(40,284)(41,285)(42,286)(43,287)(44,288)(45,289)(46,290)(47,291)(48,292)(49,293)(50,294)(51,295)(52,296)(53,297)(54,298)(55,299)(56,300)(57,301)(58,302)(59,303)(60,304)(61,229)(62,230)(63,231)(64,232)(65,233)(66,234)(67,235)(68,236)(69,237)(70,238)(71,239)(72,240)(73,241)(74,242)(75,243)(76,244)(77,182)(78,183)(79,184)(80,185)(81,186)(82,187)(83,188)(84,189)(85,190)(86,191)(87,192)(88,193)(89,194)(90,195)(91,196)(92,197)(93,198)(94,199)(95,200)(96,201)(97,202)(98,203)(99,204)(100,205)(101,206)(102,207)(103,208)(104,209)(105,210)(106,211)(107,212)(108,213)(109,214)(110,215)(111,216)(112,217)(113,218)(114,219)(115,220)(116,221)(117,222)(118,223)(119,224)(120,225)(121,226)(122,227)(123,228)(124,153)(125,154)(126,155)(127,156)(128,157)(129,158)(130,159)(131,160)(132,161)(133,162)(134,163)(135,164)(136,165)(137,166)(138,167)(139,168)(140,169)(141,170)(142,171)(143,172)(144,173)(145,174)(146,175)(147,176)(148,177)(149,178)(150,179)(151,180)(152,181), (1,183)(2,184)(3,185)(4,186)(5,187)(6,188)(7,189)(8,190)(9,191)(10,192)(11,193)(12,194)(13,195)(14,196)(15,197)(16,198)(17,199)(18,200)(19,201)(20,202)(21,203)(22,204)(23,205)(24,206)(25,207)(26,208)(27,209)(28,210)(29,211)(30,212)(31,213)(32,214)(33,215)(34,216)(35,217)(36,218)(37,219)(38,220)(39,221)(40,222)(41,223)(42,224)(43,225)(44,226)(45,227)(46,228)(47,153)(48,154)(49,155)(50,156)(51,157)(52,158)(53,159)(54,160)(55,161)(56,162)(57,163)(58,164)(59,165)(60,166)(61,167)(62,168)(63,169)(64,170)(65,171)(66,172)(67,173)(68,174)(69,175)(70,176)(71,177)(72,178)(73,179)(74,180)(75,181)(76,182)(77,244)(78,245)(79,246)(80,247)(81,248)(82,249)(83,250)(84,251)(85,252)(86,253)(87,254)(88,255)(89,256)(90,257)(91,258)(92,259)(93,260)(94,261)(95,262)(96,263)(97,264)(98,265)(99,266)(100,267)(101,268)(102,269)(103,270)(104,271)(105,272)(106,273)(107,274)(108,275)(109,276)(110,277)(111,278)(112,279)(113,280)(114,281)(115,282)(116,283)(117,284)(118,285)(119,286)(120,287)(121,288)(122,289)(123,290)(124,291)(125,292)(126,293)(127,294)(128,295)(129,296)(130,297)(131,298)(132,299)(133,300)(134,301)(135,302)(136,303)(137,304)(138,229)(139,230)(140,231)(141,232)(142,233)(143,234)(144,235)(145,236)(146,237)(147,238)(148,239)(149,240)(150,241)(151,242)(152,243), (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)(229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304) );

G=PermutationGroup([(1,245),(2,246),(3,247),(4,248),(5,249),(6,250),(7,251),(8,252),(9,253),(10,254),(11,255),(12,256),(13,257),(14,258),(15,259),(16,260),(17,261),(18,262),(19,263),(20,264),(21,265),(22,266),(23,267),(24,268),(25,269),(26,270),(27,271),(28,272),(29,273),(30,274),(31,275),(32,276),(33,277),(34,278),(35,279),(36,280),(37,281),(38,282),(39,283),(40,284),(41,285),(42,286),(43,287),(44,288),(45,289),(46,290),(47,291),(48,292),(49,293),(50,294),(51,295),(52,296),(53,297),(54,298),(55,299),(56,300),(57,301),(58,302),(59,303),(60,304),(61,229),(62,230),(63,231),(64,232),(65,233),(66,234),(67,235),(68,236),(69,237),(70,238),(71,239),(72,240),(73,241),(74,242),(75,243),(76,244),(77,182),(78,183),(79,184),(80,185),(81,186),(82,187),(83,188),(84,189),(85,190),(86,191),(87,192),(88,193),(89,194),(90,195),(91,196),(92,197),(93,198),(94,199),(95,200),(96,201),(97,202),(98,203),(99,204),(100,205),(101,206),(102,207),(103,208),(104,209),(105,210),(106,211),(107,212),(108,213),(109,214),(110,215),(111,216),(112,217),(113,218),(114,219),(115,220),(116,221),(117,222),(118,223),(119,224),(120,225),(121,226),(122,227),(123,228),(124,153),(125,154),(126,155),(127,156),(128,157),(129,158),(130,159),(131,160),(132,161),(133,162),(134,163),(135,164),(136,165),(137,166),(138,167),(139,168),(140,169),(141,170),(142,171),(143,172),(144,173),(145,174),(146,175),(147,176),(148,177),(149,178),(150,179),(151,180),(152,181)], [(1,183),(2,184),(3,185),(4,186),(5,187),(6,188),(7,189),(8,190),(9,191),(10,192),(11,193),(12,194),(13,195),(14,196),(15,197),(16,198),(17,199),(18,200),(19,201),(20,202),(21,203),(22,204),(23,205),(24,206),(25,207),(26,208),(27,209),(28,210),(29,211),(30,212),(31,213),(32,214),(33,215),(34,216),(35,217),(36,218),(37,219),(38,220),(39,221),(40,222),(41,223),(42,224),(43,225),(44,226),(45,227),(46,228),(47,153),(48,154),(49,155),(50,156),(51,157),(52,158),(53,159),(54,160),(55,161),(56,162),(57,163),(58,164),(59,165),(60,166),(61,167),(62,168),(63,169),(64,170),(65,171),(66,172),(67,173),(68,174),(69,175),(70,176),(71,177),(72,178),(73,179),(74,180),(75,181),(76,182),(77,244),(78,245),(79,246),(80,247),(81,248),(82,249),(83,250),(84,251),(85,252),(86,253),(87,254),(88,255),(89,256),(90,257),(91,258),(92,259),(93,260),(94,261),(95,262),(96,263),(97,264),(98,265),(99,266),(100,267),(101,268),(102,269),(103,270),(104,271),(105,272),(106,273),(107,274),(108,275),(109,276),(110,277),(111,278),(112,279),(113,280),(114,281),(115,282),(116,283),(117,284),(118,285),(119,286),(120,287),(121,288),(122,289),(123,290),(124,291),(125,292),(126,293),(127,294),(128,295),(129,296),(130,297),(131,298),(132,299),(133,300),(134,301),(135,302),(136,303),(137,304),(138,229),(139,230),(140,231),(141,232),(142,233),(143,234),(144,235),(145,236),(146,237),(147,238),(148,239),(149,240),(150,241),(151,242),(152,243)], [(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),(229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304)])

304 conjugacy classes

class 1 2A···2G4A···4H19A···19R38A···38DV76A···76EN
order12···24···419···1938···3876···76
size11···11···11···11···11···1

304 irreducible representations

dim11111111
type+++
imageC1C2C2C4C19C38C38C76
kernelC22×C76C2×C76C22×C38C2×C38C22×C4C2×C4C23C22
# reps16181810818144

Matrix representation of C22×C76 in GL3(𝔽229) generated by

22800
02280
00228
,
100
010
00228
,
3400
02130
00225
G:=sub<GL(3,GF(229))| [228,0,0,0,228,0,0,0,228],[1,0,0,0,1,0,0,0,228],[34,0,0,0,213,0,0,0,225] >;

C22×C76 in GAP, Magma, Sage, TeX

C_2^2\times C_{76}
% in TeX

G:=Group("C2^2xC76");
// GroupNames label

G:=SmallGroup(304,37);
// by ID

G=gap.SmallGroup(304,37);
# by ID

G:=PCGroup([5,-2,-2,-2,-19,-2,760]);
// Polycyclic

G:=Group<a,b,c|a^2=b^2=c^76=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

׿
×
𝔽