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G = D5×C23×C6  order 480 = 25·3·5

Direct product of C23×C6 and D5

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D5×C23×C6, C15⋊3C25, C30⋊3C24, C5⋊(C24×C6), C10⋊(C23×C6), (C23×C30)⋊8C2, (C23×C10)⋊13C6, (C2×C30)⋊13C23, (C22×C30)⋊22C22, (C2×C10)⋊6(C22×C6), (C22×C10)⋊10(C2×C6), SmallGroup(480,1210)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C5 — D5×C23×C6
Chief series
C1 — C5 — C15 — C3×D5 — C6×D5 — D5×C2×C6 — D5×C22×C6 — D5×C23×C6
Lower central
C5 — D5×C23×C6

Subgroups: 3952 in 1496 conjugacy classes, 882 normal (10 characteristic)
C1, C2 [×15], C2 [×16], C3, C22 [×35], C22 [×120], C5, C6 [×15], C6 [×16], C23 [×15], C23 [×140], D5 [×16], C10 [×15], C2×C6 [×35], C2×C6 [×120], C15, C24, C24 [×30], D10 [×120], C2×C10 [×35], C22×C6 [×15], C22×C6 [×140], C3×D5 [×16], C30 [×15], C25, C22×D5 [×140], C22×C10 [×15], C23×C6, C23×C6 [×30], C6×D5 [×120], C2×C30 [×35], C23×D5 [×30], C23×C10, C24×C6, D5×C2×C6 [×140], C22×C30 [×15], D5×C24, D5×C22×C6 [×30], C23×C30, D5×C23×C6

Quotients:
C1, C2 [×31], C3, C22 [×155], C6 [×31], C23 [×155], D5, C2×C6 [×155], C24 [×31], D10 [×15], C22×C6 [×155], C3×D5, C25, C22×D5 [×35], C23×C6 [×31], C6×D5 [×15], C23×D5 [×15], C24×C6, D5×C2×C6 [×35], D5×C24, D5×C22×C6 [×15], D5×C23×C6

Generators and relations
 G = < a,b,c,d,e,f | a2=b2=c2=d6=e5=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

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

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

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

(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)

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

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

Matrix representation ►G ⊆ GL5(𝔽31)

10000
01000
00100
000300
000030
,
300000
01000
003000
00010
00001
,
10000
030000
00100
00010
00001
,
10000
026000
003000
00010
00001
,
10000
01000
00100
00001
0003012
,
10000
01000
00100
000030
000300

G:=sub<GL(5,GF(31))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,30,0,0,0,0,0,30],[30,0,0,0,0,0,1,0,0,0,0,0,30,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,30,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,26,0,0,0,0,0,30,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,30,0,0,0,1,12],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,30,0,0,0,30,0] >;
 

192 conjugacy classes

class 1 2A···2O2P···2AE3A3B5A5B6A···6AD6AE···6BJ10A···10AD15A15B15C15D30A···30BH
order12···22···233556···66···610···101515151530···30
size11···15···511221···15···52···222222···2

192 irreducible representations

dim1111112222
type+++++
imageC1C2C2C3C6C6D5D10C3×D5C6×D5
kernelD5×C23×C6D5×C22×C6C23×C30D5×C24C23×D5C23×C10C23×C6C22×C6C24C23
# reps13012602230460

In GAP, Magma, Sage, TeX 

D_5\times C_2^3\times C_6
 
% in TeX
 
G:=Group("D5xC2^3xC6");
 
// GroupNames label
 
G:=SmallGroup(480,1210);
 
// by ID
 
G=gap.SmallGroup(480,1210);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^6=e^5=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
 
// generators/relations
 

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