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## G = D5×C23×C6order 480 = 25·3·5

### Direct product of C23×C6 and D5

Aliases: D5×C23×C6, C153C25, C303C24, 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
 Upper central C1 — C23×C6

Generators and relations for D5×C23×C6
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 >

Subgroups: 3952 in 1496 conjugacy classes, 882 normal (10 characteristic)
C1, C2, C2, C3, C22, C22, C5, C6, C6, C23, C23, D5, C10, C2×C6, C2×C6, C15, C24, C24, D10, C2×C10, C22×C6, C22×C6, C3×D5, C30, C25, C22×D5, C22×C10, C23×C6, C23×C6, C6×D5, C2×C30, C23×D5, C23×C10, C24×C6, D5×C2×C6, C22×C30, D5×C24, D5×C22×C6, C23×C30, D5×C23×C6
Quotients: C1, C2, C3, C22, C6, C23, D5, C2×C6, C24, D10, C22×C6, C3×D5, C25, C22×D5, C23×C6, C6×D5, C23×D5, C24×C6, D5×C2×C6, D5×C24, D5×C22×C6, D5×C23×C6

Smallest permutation representation of D5×C23×C6
On 240 points
Generators in S240
(1 239)(2 240)(3 235)(4 236)(5 237)(6 238)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 45)(14 46)(15 47)(16 48)(17 43)(18 44)(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)(7 169)(8 170)(9 171)(10 172)(11 173)(12 174)(13 165)(14 166)(15 167)(16 168)(17 163)(18 164)(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)(7 232)(8 233)(9 234)(10 229)(11 230)(12 231)(13 228)(14 223)(15 224)(16 225)(17 226)(18 227)(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 39 53 43)(2 31 40 54 44)(3 32 41 49 45)(4 33 42 50 46)(5 34 37 51 47)(6 35 38 52 48)(7 13 235 21 30)(8 14 236 22 25)(9 15 237 23 26)(10 16 238 24 27)(11 17 239 19 28)(12 18 240 20 29)(55 65 68 82 78)(56 66 69 83 73)(57 61 70 84 74)(58 62 71 79 75)(59 63 72 80 76)(60 64 67 81 77)(85 95 98 112 108)(86 96 99 113 103)(87 91 100 114 104)(88 92 101 109 105)(89 93 102 110 106)(90 94 97 111 107)(115 125 128 142 138)(116 126 129 143 133)(117 121 130 144 134)(118 122 131 139 135)(119 123 132 140 136)(120 124 127 141 137)(145 155 158 172 168)(146 156 159 173 163)(147 151 160 174 164)(148 152 161 169 165)(149 153 162 170 166)(150 154 157 171 167)(175 185 188 202 198)(176 186 189 203 193)(177 181 190 204 194)(178 182 191 199 195)(179 183 192 200 196)(180 184 187 201 197)(205 215 218 232 228)(206 216 219 233 223)(207 211 220 234 224)(208 212 221 229 225)(209 213 222 230 226)(210 214 217 231 227)
(1 193)(2 194)(3 195)(4 196)(5 197)(6 198)(7 95)(8 96)(9 91)(10 92)(11 93)(12 94)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 110)(20 111)(21 112)(22 113)(23 114)(24 109)(25 99)(26 100)(27 101)(28 102)(29 97)(30 98)(31 204)(32 199)(33 200)(34 201)(35 202)(36 203)(37 187)(38 188)(39 189)(40 190)(41 191)(42 192)(43 176)(44 177)(45 178)(46 179)(47 180)(48 175)(49 182)(50 183)(51 184)(52 185)(53 186)(54 181)(55 168)(56 163)(57 164)(58 165)(59 166)(60 167)(61 174)(62 169)(63 170)(64 171)(65 172)(66 173)(67 157)(68 158)(69 159)(70 160)(71 161)(72 162)(73 146)(74 147)(75 148)(76 149)(77 150)(78 145)(79 152)(80 153)(81 154)(82 155)(83 156)(84 151)(103 236)(104 237)(105 238)(106 239)(107 240)(108 235)(115 228)(116 223)(117 224)(118 225)(119 226)(120 227)(121 234)(122 229)(123 230)(124 231)(125 232)(126 233)(127 217)(128 218)(129 219)(130 220)(131 221)(132 222)(133 206)(134 207)(135 208)(136 209)(137 210)(138 205)(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)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,45)(14,46)(15,47)(16,48)(17,43)(18,44)(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)(7,169)(8,170)(9,171)(10,172)(11,173)(12,174)(13,165)(14,166)(15,167)(16,168)(17,163)(18,164)(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)(7,232)(8,233)(9,234)(10,229)(11,230)(12,231)(13,228)(14,223)(15,224)(16,225)(17,226)(18,227)(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), (1,36,39,53,43)(2,31,40,54,44)(3,32,41,49,45)(4,33,42,50,46)(5,34,37,51,47)(6,35,38,52,48)(7,13,235,21,30)(8,14,236,22,25)(9,15,237,23,26)(10,16,238,24,27)(11,17,239,19,28)(12,18,240,20,29)(55,65,68,82,78)(56,66,69,83,73)(57,61,70,84,74)(58,62,71,79,75)(59,63,72,80,76)(60,64,67,81,77)(85,95,98,112,108)(86,96,99,113,103)(87,91,100,114,104)(88,92,101,109,105)(89,93,102,110,106)(90,94,97,111,107)(115,125,128,142,138)(116,126,129,143,133)(117,121,130,144,134)(118,122,131,139,135)(119,123,132,140,136)(120,124,127,141,137)(145,155,158,172,168)(146,156,159,173,163)(147,151,160,174,164)(148,152,161,169,165)(149,153,162,170,166)(150,154,157,171,167)(175,185,188,202,198)(176,186,189,203,193)(177,181,190,204,194)(178,182,191,199,195)(179,183,192,200,196)(180,184,187,201,197)(205,215,218,232,228)(206,216,219,233,223)(207,211,220,234,224)(208,212,221,229,225)(209,213,222,230,226)(210,214,217,231,227), (1,193)(2,194)(3,195)(4,196)(5,197)(6,198)(7,95)(8,96)(9,91)(10,92)(11,93)(12,94)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,110)(20,111)(21,112)(22,113)(23,114)(24,109)(25,99)(26,100)(27,101)(28,102)(29,97)(30,98)(31,204)(32,199)(33,200)(34,201)(35,202)(36,203)(37,187)(38,188)(39,189)(40,190)(41,191)(42,192)(43,176)(44,177)(45,178)(46,179)(47,180)(48,175)(49,182)(50,183)(51,184)(52,185)(53,186)(54,181)(55,168)(56,163)(57,164)(58,165)(59,166)(60,167)(61,174)(62,169)(63,170)(64,171)(65,172)(66,173)(67,157)(68,158)(69,159)(70,160)(71,161)(72,162)(73,146)(74,147)(75,148)(76,149)(77,150)(78,145)(79,152)(80,153)(81,154)(82,155)(83,156)(84,151)(103,236)(104,237)(105,238)(106,239)(107,240)(108,235)(115,228)(116,223)(117,224)(118,225)(119,226)(120,227)(121,234)(122,229)(123,230)(124,231)(125,232)(126,233)(127,217)(128,218)(129,219)(130,220)(131,221)(132,222)(133,206)(134,207)(135,208)(136,209)(137,210)(138,205)(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)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,45)(14,46)(15,47)(16,48)(17,43)(18,44)(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)(7,169)(8,170)(9,171)(10,172)(11,173)(12,174)(13,165)(14,166)(15,167)(16,168)(17,163)(18,164)(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)(7,232)(8,233)(9,234)(10,229)(11,230)(12,231)(13,228)(14,223)(15,224)(16,225)(17,226)(18,227)(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), (1,36,39,53,43)(2,31,40,54,44)(3,32,41,49,45)(4,33,42,50,46)(5,34,37,51,47)(6,35,38,52,48)(7,13,235,21,30)(8,14,236,22,25)(9,15,237,23,26)(10,16,238,24,27)(11,17,239,19,28)(12,18,240,20,29)(55,65,68,82,78)(56,66,69,83,73)(57,61,70,84,74)(58,62,71,79,75)(59,63,72,80,76)(60,64,67,81,77)(85,95,98,112,108)(86,96,99,113,103)(87,91,100,114,104)(88,92,101,109,105)(89,93,102,110,106)(90,94,97,111,107)(115,125,128,142,138)(116,126,129,143,133)(117,121,130,144,134)(118,122,131,139,135)(119,123,132,140,136)(120,124,127,141,137)(145,155,158,172,168)(146,156,159,173,163)(147,151,160,174,164)(148,152,161,169,165)(149,153,162,170,166)(150,154,157,171,167)(175,185,188,202,198)(176,186,189,203,193)(177,181,190,204,194)(178,182,191,199,195)(179,183,192,200,196)(180,184,187,201,197)(205,215,218,232,228)(206,216,219,233,223)(207,211,220,234,224)(208,212,221,229,225)(209,213,222,230,226)(210,214,217,231,227), (1,193)(2,194)(3,195)(4,196)(5,197)(6,198)(7,95)(8,96)(9,91)(10,92)(11,93)(12,94)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,110)(20,111)(21,112)(22,113)(23,114)(24,109)(25,99)(26,100)(27,101)(28,102)(29,97)(30,98)(31,204)(32,199)(33,200)(34,201)(35,202)(36,203)(37,187)(38,188)(39,189)(40,190)(41,191)(42,192)(43,176)(44,177)(45,178)(46,179)(47,180)(48,175)(49,182)(50,183)(51,184)(52,185)(53,186)(54,181)(55,168)(56,163)(57,164)(58,165)(59,166)(60,167)(61,174)(62,169)(63,170)(64,171)(65,172)(66,173)(67,157)(68,158)(69,159)(70,160)(71,161)(72,162)(73,146)(74,147)(75,148)(76,149)(77,150)(78,145)(79,152)(80,153)(81,154)(82,155)(83,156)(84,151)(103,236)(104,237)(105,238)(106,239)(107,240)(108,235)(115,228)(116,223)(117,224)(118,225)(119,226)(120,227)(121,234)(122,229)(123,230)(124,231)(125,232)(126,233)(127,217)(128,218)(129,219)(130,220)(131,221)(132,222)(133,206)(134,207)(135,208)(136,209)(137,210)(138,205)(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),(7,49),(8,50),(9,51),(10,52),(11,53),(12,54),(13,45),(14,46),(15,47),(16,48),(17,43),(18,44),(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),(7,169),(8,170),(9,171),(10,172),(11,173),(12,174),(13,165),(14,166),(15,167),(16,168),(17,163),(18,164),(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),(7,232),(8,233),(9,234),(10,229),(11,230),(12,231),(13,228),(14,223),(15,224),(16,225),(17,226),(18,227),(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)], [(1,36,39,53,43),(2,31,40,54,44),(3,32,41,49,45),(4,33,42,50,46),(5,34,37,51,47),(6,35,38,52,48),(7,13,235,21,30),(8,14,236,22,25),(9,15,237,23,26),(10,16,238,24,27),(11,17,239,19,28),(12,18,240,20,29),(55,65,68,82,78),(56,66,69,83,73),(57,61,70,84,74),(58,62,71,79,75),(59,63,72,80,76),(60,64,67,81,77),(85,95,98,112,108),(86,96,99,113,103),(87,91,100,114,104),(88,92,101,109,105),(89,93,102,110,106),(90,94,97,111,107),(115,125,128,142,138),(116,126,129,143,133),(117,121,130,144,134),(118,122,131,139,135),(119,123,132,140,136),(120,124,127,141,137),(145,155,158,172,168),(146,156,159,173,163),(147,151,160,174,164),(148,152,161,169,165),(149,153,162,170,166),(150,154,157,171,167),(175,185,188,202,198),(176,186,189,203,193),(177,181,190,204,194),(178,182,191,199,195),(179,183,192,200,196),(180,184,187,201,197),(205,215,218,232,228),(206,216,219,233,223),(207,211,220,234,224),(208,212,221,229,225),(209,213,222,230,226),(210,214,217,231,227)], [(1,193),(2,194),(3,195),(4,196),(5,197),(6,198),(7,95),(8,96),(9,91),(10,92),(11,93),(12,94),(13,85),(14,86),(15,87),(16,88),(17,89),(18,90),(19,110),(20,111),(21,112),(22,113),(23,114),(24,109),(25,99),(26,100),(27,101),(28,102),(29,97),(30,98),(31,204),(32,199),(33,200),(34,201),(35,202),(36,203),(37,187),(38,188),(39,189),(40,190),(41,191),(42,192),(43,176),(44,177),(45,178),(46,179),(47,180),(48,175),(49,182),(50,183),(51,184),(52,185),(53,186),(54,181),(55,168),(56,163),(57,164),(58,165),(59,166),(60,167),(61,174),(62,169),(63,170),(64,171),(65,172),(66,173),(67,157),(68,158),(69,159),(70,160),(71,161),(72,162),(73,146),(74,147),(75,148),(76,149),(77,150),(78,145),(79,152),(80,153),(81,154),(82,155),(83,156),(84,151),(103,236),(104,237),(105,238),(106,239),(107,240),(108,235),(115,228),(116,223),(117,224),(118,225),(119,226),(120,227),(121,234),(122,229),(123,230),(124,231),(125,232),(126,233),(127,217),(128,218),(129,219),(130,220),(131,221),(132,222),(133,206),(134,207),(135,208),(136,209),(137,210),(138,205),(139,212),(140,213),(141,214),(142,215),(143,216),(144,211)]])

192 conjugacy classes

 class 1 2A ··· 2O 2P ··· 2AE 3A 3B 5A 5B 6A ··· 6AD 6AE ··· 6BJ 10A ··· 10AD 15A 15B 15C 15D 30A ··· 30BH order 1 2 ··· 2 2 ··· 2 3 3 5 5 6 ··· 6 6 ··· 6 10 ··· 10 15 15 15 15 30 ··· 30 size 1 1 ··· 1 5 ··· 5 1 1 2 2 1 ··· 1 5 ··· 5 2 ··· 2 2 2 2 2 2 ··· 2

192 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C3 C6 C6 D5 D10 C3×D5 C6×D5 kernel D5×C23×C6 D5×C22×C6 C23×C30 D5×C24 C23×D5 C23×C10 C23×C6 C22×C6 C24 C23 # reps 1 30 1 2 60 2 2 30 4 60

Matrix representation of D5×C23×C6 in GL5(𝔽31)

 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 1 0 0 0 30 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

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] >;

D5×C23×C6 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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