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## G = C23×C34order 272 = 24·17

### Abelian group of type [2,2,2,34]

Aliases: C23×C34, SmallGroup(272,54)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23×C34
 Chief series C1 — C17 — C34 — C2×C34 — C22×C34 — C23×C34
 Lower central C1 — C23×C34
 Upper central C1 — C23×C34

Generators and relations for C23×C34
G = < a,b,c,d | a2=b2=c2=d34=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >

Subgroups: 134, all normal (4 characteristic)
C1, C2, C22, C23, C24, C17, C34, C2×C34, C22×C34, C23×C34
Quotients: C1, C2, C22, C23, C24, C17, C34, C2×C34, C22×C34, C23×C34

Smallest permutation representation of C23×C34
Regular action on 272 points
Generators in S272
(1 161)(2 162)(3 163)(4 164)(5 165)(6 166)(7 167)(8 168)(9 169)(10 170)(11 137)(12 138)(13 139)(14 140)(15 141)(16 142)(17 143)(18 144)(19 145)(20 146)(21 147)(22 148)(23 149)(24 150)(25 151)(26 152)(27 153)(28 154)(29 155)(30 156)(31 157)(32 158)(33 159)(34 160)(35 173)(36 174)(37 175)(38 176)(39 177)(40 178)(41 179)(42 180)(43 181)(44 182)(45 183)(46 184)(47 185)(48 186)(49 187)(50 188)(51 189)(52 190)(53 191)(54 192)(55 193)(56 194)(57 195)(58 196)(59 197)(60 198)(61 199)(62 200)(63 201)(64 202)(65 203)(66 204)(67 171)(68 172)(69 234)(70 235)(71 236)(72 237)(73 238)(74 205)(75 206)(76 207)(77 208)(78 209)(79 210)(80 211)(81 212)(82 213)(83 214)(84 215)(85 216)(86 217)(87 218)(88 219)(89 220)(90 221)(91 222)(92 223)(93 224)(94 225)(95 226)(96 227)(97 228)(98 229)(99 230)(100 231)(101 232)(102 233)(103 243)(104 244)(105 245)(106 246)(107 247)(108 248)(109 249)(110 250)(111 251)(112 252)(113 253)(114 254)(115 255)(116 256)(117 257)(118 258)(119 259)(120 260)(121 261)(122 262)(123 263)(124 264)(125 265)(126 266)(127 267)(128 268)(129 269)(130 270)(131 271)(132 272)(133 239)(134 240)(135 241)(136 242)
(1 99)(2 100)(3 101)(4 102)(5 69)(6 70)(7 71)(8 72)(9 73)(10 74)(11 75)(12 76)(13 77)(14 78)(15 79)(16 80)(17 81)(18 82)(19 83)(20 84)(21 85)(22 86)(23 87)(24 88)(25 89)(26 90)(27 91)(28 92)(29 93)(30 94)(31 95)(32 96)(33 97)(34 98)(35 131)(36 132)(37 133)(38 134)(39 135)(40 136)(41 103)(42 104)(43 105)(44 106)(45 107)(46 108)(47 109)(48 110)(49 111)(50 112)(51 113)(52 114)(53 115)(54 116)(55 117)(56 118)(57 119)(58 120)(59 121)(60 122)(61 123)(62 124)(63 125)(64 126)(65 127)(66 128)(67 129)(68 130)(137 206)(138 207)(139 208)(140 209)(141 210)(142 211)(143 212)(144 213)(145 214)(146 215)(147 216)(148 217)(149 218)(150 219)(151 220)(152 221)(153 222)(154 223)(155 224)(156 225)(157 226)(158 227)(159 228)(160 229)(161 230)(162 231)(163 232)(164 233)(165 234)(166 235)(167 236)(168 237)(169 238)(170 205)(171 269)(172 270)(173 271)(174 272)(175 239)(176 240)(177 241)(178 242)(179 243)(180 244)(181 245)(182 246)(183 247)(184 248)(185 249)(186 250)(187 251)(188 252)(189 253)(190 254)(191 255)(192 256)(193 257)(194 258)(195 259)(196 260)(197 261)(198 262)(199 263)(200 264)(201 265)(202 266)(203 267)(204 268)
(1 63)(2 64)(3 65)(4 66)(5 67)(6 68)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 61)(34 62)(69 129)(70 130)(71 131)(72 132)(73 133)(74 134)(75 135)(76 136)(77 103)(78 104)(79 105)(80 106)(81 107)(82 108)(83 109)(84 110)(85 111)(86 112)(87 113)(88 114)(89 115)(90 116)(91 117)(92 118)(93 119)(94 120)(95 121)(96 122)(97 123)(98 124)(99 125)(100 126)(101 127)(102 128)(137 177)(138 178)(139 179)(140 180)(141 181)(142 182)(143 183)(144 184)(145 185)(146 186)(147 187)(148 188)(149 189)(150 190)(151 191)(152 192)(153 193)(154 194)(155 195)(156 196)(157 197)(158 198)(159 199)(160 200)(161 201)(162 202)(163 203)(164 204)(165 171)(166 172)(167 173)(168 174)(169 175)(170 176)(205 240)(206 241)(207 242)(208 243)(209 244)(210 245)(211 246)(212 247)(213 248)(214 249)(215 250)(216 251)(217 252)(218 253)(219 254)(220 255)(221 256)(222 257)(223 258)(224 259)(225 260)(226 261)(227 262)(228 263)(229 264)(230 265)(231 266)(232 267)(233 268)(234 269)(235 270)(236 271)(237 272)(238 239)
(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)

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

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

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

272 conjugacy classes

 class 1 2A ··· 2O 17A ··· 17P 34A ··· 34IF order 1 2 ··· 2 17 ··· 17 34 ··· 34 size 1 1 ··· 1 1 ··· 1 1 ··· 1

272 irreducible representations

 dim 1 1 1 1 type + + image C1 C2 C17 C34 kernel C23×C34 C22×C34 C24 C23 # reps 1 15 16 240

Matrix representation of C23×C34 in GL4(𝔽103) generated by

 1 0 0 0 0 1 0 0 0 0 102 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 102 0 0 0 0 102
,
 102 0 0 0 0 1 0 0 0 0 1 0 0 0 0 102
,
 22 0 0 0 0 90 0 0 0 0 79 0 0 0 0 8
G:=sub<GL(4,GF(103))| [1,0,0,0,0,1,0,0,0,0,102,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,102,0,0,0,0,102],[102,0,0,0,0,1,0,0,0,0,1,0,0,0,0,102],[22,0,0,0,0,90,0,0,0,0,79,0,0,0,0,8] >;

C23×C34 in GAP, Magma, Sage, TeX

C_2^3\times C_{34}
% in TeX

G:=Group("C2^3xC34");
// GroupNames label

G:=SmallGroup(272,54);
// by ID

G=gap.SmallGroup(272,54);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-17]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^2=d^34=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,c*d=d*c>;
// generators/relations

׿
×
𝔽