C2^4xD15 - GroupNames

G = C₂⁴×D₁₅ order 480 = 2⁵·3·5

Direct product of C₂⁴ and D₁₅

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

Aliases: C₂⁴×D₁₅, C₁₅⋊₂C₂⁵, C₃₀⋊₂C₂⁴, C₅⋊₂(S₃×C₂⁴), C₃⋊₂(D₅×C₂⁴), C₆⋊₂(C₂³×D₅), (C₂³×C₆)⋊₇D₅, (C₂³×C₃₀)⋊₅C₂, C₁₀⋊₂(S₃×C₂³), (C₂³×C₁₀)⋊₁₁S₃, (C₂×C₃₀)⋊₁₁C₂³, (C₂²×C₆)⋊₁₅D₁₀, (C₂²×C₁₀)⋊₁₈D₆, (C₂²×C₃₀)⋊₂₀C₂², (C₂×C₆)⋊₁₀(C₂²×D₅), (C₂×C₁₀)⋊₁₃(C₂²×S₃), SmallGroup(480,1212)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₅ — C₂⁴×D₁₅

Chief series

C₁ — C₅ — C₁₅ — D₁₅ — D₃₀ — C₂²×D₁₅ — C₂³×D₁₅ — C₂⁴×D₁₅

Lower central

C₁₅ — C₂⁴×D₁₅

Upper central

C₁ — C₂⁴

Generators and relations for C₂⁴×D₁₅
G = < a,b,c,d,e,f | a²=b²=c²=d²=e¹⁵=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: 7636 in 1496 conjugacy classes, 575 normal (9 characteristic)
C₁, C₂, C₂, C₃, C₂², C₂², C₅, S₃, C₆, C₂³, C₂³, D₅, C₁₀, D₆, C₂×C₆, C₁₅, C₂⁴, C₂⁴, D₁₀, C₂×C₁₀, C₂²×S₃, C₂²×C₆, D₁₅, C₃₀, C₂⁵, C₂²×D₅, C₂²×C₁₀, S₃×C₂³, C₂³×C₆, D₃₀, C₂×C₃₀, C₂³×D₅, C₂³×C₁₀, S₃×C₂⁴, C₂²×D₁₅, C₂²×C₃₀, D₅×C₂⁴, C₂³×D₁₅, C₂³×C₃₀, C₂⁴×D₁₅
Quotients: C₁, C₂, C₂², S₃, C₂³, D₅, D₆, C₂⁴, D₁₀, C₂²×S₃, D₁₅, C₂⁵, C₂²×D₅, S₃×C₂³, D₃₀, C₂³×D₅, S₃×C₂⁴, C₂²×D₁₅, D₅×C₂⁴, C₂³×D₁₅, C₂⁴×D₁₅

Smallest permutation representation of C₂⁴×D₁₅
►On 240 points

Generators in S₂₄₀

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

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

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

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

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

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

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

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

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

144 conjugacy classes

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

144 irreducible representations

dim	1	1	1	2	2	2	2	2	2
type	+	+	+	+	+	+	+	+	+
image	C₁	C₂	C₂	S₃	D₅	D₆	D₁₀	D₁₅	D₃₀
kernel	C₂⁴×D₁₅	C₂³×D₁₅	C₂³×C₃₀	C₂³×C₁₀	C₂³×C₆	C₂²×C₁₀	C₂²×C₆	C₂⁴	C₂³
# reps	1	30	1	1	2	15	30	4	60

Matrix representation of C₂⁴×D₁₅ ►in GL₆(𝔽₃₁)

30	0	0	0	0	0
0	30	0	0	0	0
0	0	30	0	0	0
0	0	0	30	0	0
0	0	0	0	1	0
0	0	0	0	0	1

30	0	0	0	0	0
0	1	0	0	0	0
0	0	30	0	0	0
0	0	0	30	0	0
0	0	0	0	30	0
0	0	0	0	0	30

1	0	0	0	0	0
0	30	0	0	0	0
0	0	30	0	0	0
0	0	0	30	0	0
0	0	0	0	30	0
0	0	0	0	0	30

30	0	0	0	0	0
0	30	0	0	0	0
0	0	30	0	0	0
0	0	0	30	0	0
0	0	0	0	30	0
0	0	0	0	0	30

1	0	0	0	0	0
0	1	0	0	0	0
0	0	21	21	0	0
0	0	6	9	0	0
0	0	0	0	23	11
0	0	0	0	20	15

30	0	0	0	0	0
0	30	0	0	0	0
0	0	10	10	0	0
0	0	18	21	0	0
0	0	0	0	5	14
0	0	0	0	16	26

G:=sub<GL(6,GF(31))| [30,0,0,0,0,0,0,30,0,0,0,0,0,0,30,0,0,0,0,0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[30,0,0,0,0,0,0,1,0,0,0,0,0,0,30,0,0,0,0,0,0,30,0,0,0,0,0,0,30,0,0,0,0,0,0,30],[1,0,0,0,0,0,0,30,0,0,0,0,0,0,30,0,0,0,0,0,0,30,0,0,0,0,0,0,30,0,0,0,0,0,0,30],[30,0,0,0,0,0,0,30,0,0,0,0,0,0,30,0,0,0,0,0,0,30,0,0,0,0,0,0,30,0,0,0,0,0,0,30],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,21,6,0,0,0,0,21,9,0,0,0,0,0,0,23,20,0,0,0,0,11,15],[30,0,0,0,0,0,0,30,0,0,0,0,0,0,10,18,0,0,0,0,10,21,0,0,0,0,0,0,5,16,0,0,0,0,14,26] >;

C₂⁴×D₁₅ in GAP, Magma, Sage, TeX

C_2^4\times D_{15}

% in TeX

G:=Group("C2^4xD15");

// GroupNames label

G:=SmallGroup(480,1212);

// by ID

G=gap.SmallGroup(480,1212);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,2693,18822]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^15=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

G = C24×D15 order 480 = 25·3·5

Direct product of C24 and D15

G = C₂⁴×D₁₅ order 480 = 2⁵·3·5

Direct product of C₂⁴ and D₁₅