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G = C35order 243 = 35

Elementary abelian group of type [3,3,3,3,3]

direct product, p-group, elementary abelian, monomial

Aliases: C35, SmallGroup(243,67)

Series: Derived Chief Lower central Upper central Jennings

C1 — C35
C1C3C32C33C34 — C35
C1 — C35
C1 — C35
C1 — C35

Generators and relations for C35
 G = < a,b,c,d,e | a3=b3=c3=d3=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, de=ed >

Subgroups: 2664, all normal (2 characteristic)
C1, C3 [×121], C32 [×1210], C33 [×1210], C34 [×121], C35
Quotients: C1, C3 [×121], C32 [×1210], C33 [×1210], C34 [×121], C35

Smallest permutation representation of C35
Regular action on 243 points
Generators in S243
(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)
(1 23 14)(2 24 15)(3 22 13)(4 183 174)(5 181 172)(6 182 173)(7 25 16)(8 26 17)(9 27 18)(10 28 19)(11 29 20)(12 30 21)(31 49 40)(32 50 41)(33 51 42)(34 52 43)(35 53 44)(36 54 45)(37 55 46)(38 56 47)(39 57 48)(58 76 67)(59 77 68)(60 78 69)(61 79 70)(62 80 71)(63 81 72)(64 83 73)(65 84 74)(66 82 75)(85 103 94)(86 104 95)(87 105 96)(88 106 97)(89 107 98)(90 108 99)(91 109 100)(92 110 101)(93 111 102)(112 130 121)(113 131 122)(114 132 123)(115 133 124)(116 134 125)(117 135 126)(118 136 127)(119 137 128)(120 138 129)(139 157 148)(140 158 149)(141 159 150)(142 160 151)(143 161 152)(144 162 153)(145 164 154)(146 165 155)(147 163 156)(166 184 175)(167 185 176)(168 186 177)(169 187 178)(170 188 179)(171 189 180)(190 209 200)(191 210 201)(192 208 199)(193 211 202)(194 212 203)(195 213 204)(196 214 205)(197 215 206)(198 216 207)(217 235 226)(218 236 227)(219 237 228)(220 238 229)(221 239 230)(222 240 231)(223 241 232)(224 242 233)(225 243 234)
(1 11 8)(2 12 9)(3 10 7)(4 189 186)(5 187 184)(6 188 185)(13 19 16)(14 20 17)(15 21 18)(22 28 25)(23 29 26)(24 30 27)(31 37 34)(32 38 35)(33 39 36)(40 46 43)(41 47 44)(42 48 45)(49 55 52)(50 56 53)(51 57 54)(58 64 61)(59 65 62)(60 66 63)(67 73 70)(68 74 71)(69 75 72)(76 83 79)(77 84 80)(78 82 81)(85 91 88)(86 92 89)(87 93 90)(94 100 97)(95 101 98)(96 102 99)(103 109 106)(104 110 107)(105 111 108)(112 118 115)(113 119 116)(114 120 117)(121 127 124)(122 128 125)(123 129 126)(130 136 133)(131 137 134)(132 138 135)(139 145 142)(140 146 143)(141 147 144)(148 154 151)(149 155 152)(150 156 153)(157 164 160)(158 165 161)(159 163 162)(166 172 169)(167 173 170)(168 174 171)(175 181 178)(176 182 179)(177 183 180)(190 215 212)(191 216 213)(192 214 211)(193 199 196)(194 200 197)(195 201 198)(202 208 205)(203 209 206)(204 210 207)(217 223 220)(218 224 221)(219 225 222)(226 232 229)(227 233 230)(228 234 231)(235 241 238)(236 242 239)(237 243 240)
(1 167 86)(2 168 87)(3 166 85)(4 111 30)(5 109 28)(6 110 29)(7 169 88)(8 170 89)(9 171 90)(10 172 91)(11 173 92)(12 174 93)(13 175 94)(14 176 95)(15 177 96)(16 178 97)(17 179 98)(18 180 99)(19 181 100)(20 182 101)(21 183 102)(22 184 103)(23 185 104)(24 186 105)(25 187 106)(26 188 107)(27 189 108)(31 193 112)(32 194 113)(33 195 114)(34 196 115)(35 197 116)(36 198 117)(37 199 118)(38 200 119)(39 201 120)(40 202 121)(41 203 122)(42 204 123)(43 205 124)(44 206 125)(45 207 126)(46 208 127)(47 209 128)(48 210 129)(49 211 130)(50 212 131)(51 213 132)(52 214 133)(53 215 134)(54 216 135)(55 192 136)(56 190 137)(57 191 138)(58 217 139)(59 218 140)(60 219 141)(61 220 142)(62 221 143)(63 222 144)(64 223 145)(65 224 146)(66 225 147)(67 226 148)(68 227 149)(69 228 150)(70 229 151)(71 230 152)(72 231 153)(73 232 154)(74 233 155)(75 234 156)(76 235 157)(77 236 158)(78 237 159)(79 238 160)(80 239 161)(81 240 162)(82 243 163)(83 241 164)(84 242 165)
(1 218 32)(2 219 33)(3 217 31)(4 163 191)(5 164 192)(6 165 190)(7 220 34)(8 221 35)(9 222 36)(10 223 37)(11 224 38)(12 225 39)(13 226 40)(14 227 41)(15 228 42)(16 229 43)(17 230 44)(18 231 45)(19 232 46)(20 233 47)(21 234 48)(22 235 49)(23 236 50)(24 237 51)(25 238 52)(26 239 53)(27 240 54)(28 241 55)(29 242 56)(30 243 57)(58 112 85)(59 113 86)(60 114 87)(61 115 88)(62 116 89)(63 117 90)(64 118 91)(65 119 92)(66 120 93)(67 121 94)(68 122 95)(69 123 96)(70 124 97)(71 125 98)(72 126 99)(73 127 100)(74 128 101)(75 129 102)(76 130 103)(77 131 104)(78 132 105)(79 133 106)(80 134 107)(81 135 108)(82 138 111)(83 136 109)(84 137 110)(139 193 166)(140 194 167)(141 195 168)(142 196 169)(143 197 170)(144 198 171)(145 199 172)(146 200 173)(147 201 174)(148 202 175)(149 203 176)(150 204 177)(151 205 178)(152 206 179)(153 207 180)(154 208 181)(155 209 182)(156 210 183)(157 211 184)(158 212 185)(159 213 186)(160 214 187)(161 215 188)(162 216 189)

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

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

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

C35 is a maximal subgroup of   C35⋊C2

243 conjugacy classes

class 1 3A···3IH
order13···3
size11···1

243 irreducible representations

dim11
type+
imageC1C3
kernelC35C34
# reps1242

Matrix representation of C35 in GL5(𝔽7)

10000
02000
00400
00020
00002
,
10000
02000
00400
00010
00002
,
40000
04000
00400
00040
00004
,
40000
04000
00400
00040
00002
,
10000
04000
00400
00040
00004

G:=sub<GL(5,GF(7))| [1,0,0,0,0,0,2,0,0,0,0,0,4,0,0,0,0,0,2,0,0,0,0,0,2],[1,0,0,0,0,0,2,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,2],[4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,2],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4] >;

C35 in GAP, Magma, Sage, TeX

C_3^5
% in TeX

G:=Group("C3^5");
// GroupNames label

G:=SmallGroup(243,67);
// by ID

G=gap.SmallGroup(243,67);
# by ID

G:=PCGroup([5,-3,3,3,3,3]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^3=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,d*e=e*d>;
// generators/relations

׿
×
𝔽