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## G = C3×C23.2F5order 480 = 25·3·5

### Direct product of C3 and C23.2F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×C23.2F5
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C6×Dic5 — C6×C5⋊C8 — C3×C23.2F5
 Lower central C5 — C10 — C3×C23.2F5
 Upper central C1 — C2×C6 — C22×C6

Generators and relations for C3×C23.2F5
G = < a,b,c,d,e,f | a3=b2=c2=d2=e5=1, f4=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e3 >

Subgroups: 280 in 100 conjugacy classes, 48 normal (32 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C8, C2×C4, C23, C10, C10, C12, C2×C6, C2×C6, C2×C6, C15, C2×C8, C22×C4, Dic5, Dic5, C2×C10, C2×C10, C2×C10, C24, C2×C12, C22×C6, C30, C30, C22⋊C8, C5⋊C8, C2×Dic5, C2×Dic5, C22×C10, C2×C24, C22×C12, C3×Dic5, C3×Dic5, C2×C30, C2×C30, C2×C30, C2×C5⋊C8, C22×Dic5, C3×C22⋊C8, C3×C5⋊C8, C6×Dic5, C6×Dic5, C22×C30, C23.2F5, C6×C5⋊C8, C2×C6×Dic5, C3×C23.2F5
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, D4, C12, C2×C6, C22⋊C4, C2×C8, M4(2), F5, C24, C2×C12, C3×D4, C22⋊C8, C5⋊C8, C2×F5, C3×C22⋊C4, C2×C24, C3×M4(2), C3×F5, C2×C5⋊C8, C22.F5, C22⋊F5, C3×C22⋊C8, C3×C5⋊C8, C6×F5, C23.2F5, C6×C5⋊C8, C3×C22.F5, C3×C22⋊F5, C3×C23.2F5

Smallest permutation representation of C3×C23.2F5
On 240 points
Generators in S240
(1 12 75)(2 13 76)(3 14 77)(4 15 78)(5 16 79)(6 9 80)(7 10 73)(8 11 74)(17 224 208)(18 217 201)(19 218 202)(20 219 203)(21 220 204)(22 221 205)(23 222 206)(24 223 207)(25 94 111)(26 95 112)(27 96 105)(28 89 106)(29 90 107)(30 91 108)(31 92 109)(32 93 110)(33 52 69)(34 53 70)(35 54 71)(36 55 72)(37 56 65)(38 49 66)(39 50 67)(40 51 68)(41 185 172)(42 186 173)(43 187 174)(44 188 175)(45 189 176)(46 190 169)(47 191 170)(48 192 171)(57 157 197)(58 158 198)(59 159 199)(60 160 200)(61 153 193)(62 154 194)(63 155 195)(64 156 196)(81 138 212)(82 139 213)(83 140 214)(84 141 215)(85 142 216)(86 143 209)(87 144 210)(88 137 211)(97 151 237)(98 152 238)(99 145 239)(100 146 240)(101 147 233)(102 148 234)(103 149 235)(104 150 236)(113 225 122)(114 226 123)(115 227 124)(116 228 125)(117 229 126)(118 230 127)(119 231 128)(120 232 121)(129 182 166)(130 183 167)(131 184 168)(132 177 161)(133 178 162)(134 179 163)(135 180 164)(136 181 165)
(2 202)(4 204)(6 206)(8 208)(9 23)(11 17)(13 19)(15 21)(26 192)(28 186)(30 188)(32 190)(34 143)(36 137)(38 139)(40 141)(42 106)(44 108)(46 110)(48 112)(49 213)(51 215)(53 209)(55 211)(58 236)(60 238)(62 240)(64 234)(66 82)(68 84)(70 86)(72 88)(74 224)(76 218)(78 220)(80 222)(89 173)(91 175)(93 169)(95 171)(98 160)(100 154)(102 156)(104 158)(114 168)(116 162)(118 164)(120 166)(121 182)(123 184)(125 178)(127 180)(129 232)(131 226)(133 228)(135 230)(146 194)(148 196)(150 198)(152 200)
(1 201)(2 202)(3 203)(4 204)(5 205)(6 206)(7 207)(8 208)(9 23)(10 24)(11 17)(12 18)(13 19)(14 20)(15 21)(16 22)(25 191)(26 192)(27 185)(28 186)(29 187)(30 188)(31 189)(32 190)(33 142)(34 143)(35 144)(36 137)(37 138)(38 139)(39 140)(40 141)(41 105)(42 106)(43 107)(44 108)(45 109)(46 110)(47 111)(48 112)(49 213)(50 214)(51 215)(52 216)(53 209)(54 210)(55 211)(56 212)(57 235)(58 236)(59 237)(60 238)(61 239)(62 240)(63 233)(64 234)(65 81)(66 82)(67 83)(68 84)(69 85)(70 86)(71 87)(72 88)(73 223)(74 224)(75 217)(76 218)(77 219)(78 220)(79 221)(80 222)(89 173)(90 174)(91 175)(92 176)(93 169)(94 170)(95 171)(96 172)(97 159)(98 160)(99 153)(100 154)(101 155)(102 156)(103 157)(104 158)(113 167)(114 168)(115 161)(116 162)(117 163)(118 164)(119 165)(120 166)(121 182)(122 183)(123 184)(124 177)(125 178)(126 179)(127 180)(128 181)(129 232)(130 225)(131 226)(132 227)(133 228)(134 229)(135 230)(136 231)(145 193)(146 194)(147 195)(148 196)(149 197)(150 198)(151 199)(152 200)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)(113 117)(114 118)(115 119)(116 120)(121 125)(122 126)(123 127)(124 128)(129 133)(130 134)(131 135)(132 136)(137 141)(138 142)(139 143)(140 144)(145 149)(146 150)(147 151)(148 152)(153 157)(154 158)(155 159)(156 160)(161 165)(162 166)(163 167)(164 168)(169 173)(170 174)(171 175)(172 176)(177 181)(178 182)(179 183)(180 184)(185 189)(186 190)(187 191)(188 192)(193 197)(194 198)(195 199)(196 200)(201 205)(202 206)(203 207)(204 208)(209 213)(210 214)(211 215)(212 216)(217 221)(218 222)(219 223)(220 224)(225 229)(226 230)(227 231)(228 232)(233 237)(234 238)(235 239)(236 240)
(1 153 107 81 132)(2 82 154 133 108)(3 134 83 109 155)(4 110 135 156 84)(5 157 111 85 136)(6 86 158 129 112)(7 130 87 105 159)(8 106 131 160 88)(9 143 198 182 26)(10 183 144 27 199)(11 28 184 200 137)(12 193 29 138 177)(13 139 194 178 30)(14 179 140 31 195)(15 32 180 196 141)(16 197 25 142 181)(17 186 123 152 36)(18 145 187 37 124)(19 38 146 125 188)(20 126 39 189 147)(21 190 127 148 40)(22 149 191 33 128)(23 34 150 121 192)(24 122 35 185 151)(41 97 207 225 71)(42 226 98 72 208)(43 65 227 201 99)(44 202 66 100 228)(45 101 203 229 67)(46 230 102 68 204)(47 69 231 205 103)(48 206 70 104 232)(49 240 116 175 218)(50 176 233 219 117)(51 220 169 118 234)(52 119 221 235 170)(53 236 120 171 222)(54 172 237 223 113)(55 224 173 114 238)(56 115 217 239 174)(57 94 216 165 79)(58 166 95 80 209)(59 73 167 210 96)(60 211 74 89 168)(61 90 212 161 75)(62 162 91 76 213)(63 77 163 214 92)(64 215 78 93 164)
(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)

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 4E 4F 5 6A ··· 6F 6G 6H 6I 6J 8A ··· 8H 10A ··· 10G 12A ··· 12H 12I 12J 12K 12L 15A 15B 24A ··· 24P 30A ··· 30N order 1 2 2 2 2 2 3 3 4 4 4 4 4 4 5 6 ··· 6 6 6 6 6 8 ··· 8 10 ··· 10 12 ··· 12 12 12 12 12 15 15 24 ··· 24 30 ··· 30 size 1 1 1 1 2 2 1 1 5 5 5 5 10 10 4 1 ··· 1 2 2 2 2 10 ··· 10 4 ··· 4 5 ··· 5 10 10 10 10 4 4 10 ··· 10 4 ··· 4

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 4 4 4 4 4 4 4 type + + + + + - + - + image C1 C2 C2 C3 C4 C4 C6 C6 C8 C12 C12 C24 D4 M4(2) C3×D4 C3×M4(2) F5 C5⋊C8 C2×F5 C3×F5 C22.F5 C22⋊F5 C3×C5⋊C8 C6×F5 C3×C22.F5 C3×C22⋊F5 kernel C3×C23.2F5 C6×C5⋊C8 C2×C6×Dic5 C23.2F5 C6×Dic5 C22×C30 C2×C5⋊C8 C22×Dic5 C2×C30 C2×Dic5 C22×C10 C2×C10 C3×Dic5 C30 Dic5 C10 C22×C6 C2×C6 C2×C6 C23 C6 C6 C22 C22 C2 C2 # reps 1 2 1 2 2 2 4 2 8 4 4 16 2 2 4 4 1 2 1 2 2 2 4 2 4 4

Matrix representation of C3×C23.2F5 in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 225 0 0 0 0 0 0 225 0 0 0 0 0 0 225 0 0 0 0 0 0 225
,
 1 0 0 0 0 0 23 240 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 140 208 240 0 0 0 220 17 0 240
,
 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 51 1 0 0 0 0 240 0 0 0 0 0 221 0 190 51 0 0 3 0 190 240
,
 23 239 0 0 0 0 24 218 0 0 0 0 0 0 140 208 239 0 0 0 220 17 0 239 0 0 26 140 101 33 0 0 162 128 21 224

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,225,0,0,0,0,0,0,225,0,0,0,0,0,0,225,0,0,0,0,0,0,225],[1,23,0,0,0,0,0,240,0,0,0,0,0,0,1,0,140,220,0,0,0,1,208,17,0,0,0,0,240,0,0,0,0,0,0,240],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,51,240,221,3,0,0,1,0,0,0,0,0,0,0,190,190,0,0,0,0,51,240],[23,24,0,0,0,0,239,218,0,0,0,0,0,0,140,220,26,162,0,0,208,17,140,128,0,0,239,0,101,21,0,0,0,239,33,224] >;

C3×C23.2F5 in GAP, Magma, Sage, TeX

C_3\times C_2^3._2F_5
% in TeX

G:=Group("C3xC2^3.2F5");
// GroupNames label

G:=SmallGroup(480,292);
// by ID

G=gap.SmallGroup(480,292);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,365,136,9414,1595]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=e^5=1,f^4=d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,f*b*f^-1=b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^3>;
// generators/relations

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