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G = C24×D15order 480 = 25·3·5

Direct product of C24 and D15

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

Aliases: C24×D15, C152C25, C302C24, C52(S3×C24), C32(D5×C24), C62(C23×D5), (C23×C6)⋊7D5, (C23×C30)⋊5C2, C102(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

C1C15 — C24×D15
C1C5C15D15D30C22×D15C23×D15 — C24×D15
C15 — C24×D15
C1C24

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 [×15], C2 [×16], C3, C22 [×35], C22 [×120], C5, S3 [×16], C6 [×15], C23 [×15], C23 [×140], D5 [×16], C10 [×15], D6 [×120], C2×C6 [×35], C15, C24, C24 [×30], D10 [×120], C2×C10 [×35], C22×S3 [×140], C22×C6 [×15], D15 [×16], C30 [×15], C25, C22×D5 [×140], C22×C10 [×15], S3×C23 [×30], C23×C6, D30 [×120], C2×C30 [×35], C23×D5 [×30], C23×C10, S3×C24, C22×D15 [×140], C22×C30 [×15], D5×C24, C23×D15 [×30], C23×C30, C24×D15
Quotients: C1, C2 [×31], C22 [×155], S3, C23 [×155], D5, D6 [×15], C24 [×31], D10 [×15], C22×S3 [×35], D15, C25, C22×D5 [×35], S3×C23 [×15], D30 [×15], C23×D5 [×15], S3×C24, C22×D15 [×35], D5×C24, C23×D15 [×15], C24×D15

Smallest permutation representation of C24×D15
On 240 points
Generators in S240
(1 234)(2 235)(3 236)(4 237)(5 238)(6 239)(7 240)(8 226)(9 227)(10 228)(11 229)(12 230)(13 231)(14 232)(15 233)(16 214)(17 215)(18 216)(19 217)(20 218)(21 219)(22 220)(23 221)(24 222)(25 223)(26 224)(27 225)(28 211)(29 212)(30 213)(31 199)(32 200)(33 201)(34 202)(35 203)(36 204)(37 205)(38 206)(39 207)(40 208)(41 209)(42 210)(43 196)(44 197)(45 198)(46 181)(47 182)(48 183)(49 184)(50 185)(51 186)(52 187)(53 188)(54 189)(55 190)(56 191)(57 192)(58 193)(59 194)(60 195)(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)(76 162)(77 163)(78 164)(79 165)(80 151)(81 152)(82 153)(83 154)(84 155)(85 156)(86 157)(87 158)(88 159)(89 160)(90 161)(91 148)(92 149)(93 150)(94 136)(95 137)(96 138)(97 139)(98 140)(99 141)(100 142)(101 143)(102 144)(103 145)(104 146)(105 147)(106 125)(107 126)(108 127)(109 128)(110 129)(111 130)(112 131)(113 132)(114 133)(115 134)(116 135)(117 121)(118 122)(119 123)(120 124)
(1 69)(2 70)(3 71)(4 72)(5 73)(6 74)(7 75)(8 61)(9 62)(10 63)(11 64)(12 65)(13 66)(14 67)(15 68)(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 94)(32 95)(33 96)(34 97)(35 98)(36 99)(37 100)(38 101)(39 102)(40 103)(41 104)(42 105)(43 91)(44 92)(45 93)(46 113)(47 114)(48 115)(49 116)(50 117)(51 118)(52 119)(53 120)(54 106)(55 107)(56 108)(57 109)(58 110)(59 111)(60 112)(121 185)(122 186)(123 187)(124 188)(125 189)(126 190)(127 191)(128 192)(129 193)(130 194)(131 195)(132 181)(133 182)(134 183)(135 184)(136 199)(137 200)(138 201)(139 202)(140 203)(141 204)(142 205)(143 206)(144 207)(145 208)(146 209)(147 210)(148 196)(149 197)(150 198)(151 211)(152 212)(153 213)(154 214)(155 215)(156 216)(157 217)(158 218)(159 219)(160 220)(161 221)(162 222)(163 223)(164 224)(165 225)(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 32)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 43)(13 44)(14 45)(15 31)(16 57)(17 58)(18 59)(19 60)(20 46)(21 47)(22 48)(23 49)(24 50)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)(61 102)(62 103)(63 104)(64 105)(65 91)(66 92)(67 93)(68 94)(69 95)(70 96)(71 97)(72 98)(73 99)(74 100)(75 101)(76 117)(77 118)(78 119)(79 120)(80 106)(81 107)(82 108)(83 109)(84 110)(85 111)(86 112)(87 113)(88 114)(89 115)(90 116)(121 162)(122 163)(123 164)(124 165)(125 151)(126 152)(127 153)(128 154)(129 155)(130 156)(131 157)(132 158)(133 159)(134 160)(135 161)(136 173)(137 174)(138 175)(139 176)(140 177)(141 178)(142 179)(143 180)(144 166)(145 167)(146 168)(147 169)(148 170)(149 171)(150 172)(181 218)(182 219)(183 220)(184 221)(185 222)(186 223)(187 224)(188 225)(189 211)(190 212)(191 213)(192 214)(193 215)(194 216)(195 217)(196 230)(197 231)(198 232)(199 233)(200 234)(201 235)(202 236)(203 237)(204 238)(205 239)(206 240)(207 226)(208 227)(209 228)(210 229)
(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 57)(32 58)(33 59)(34 60)(35 46)(36 47)(37 48)(38 49)(39 50)(40 51)(41 52)(42 53)(43 54)(44 55)(45 56)(61 76)(62 77)(63 78)(64 79)(65 80)(66 81)(67 82)(68 83)(69 84)(70 85)(71 86)(72 87)(73 88)(74 89)(75 90)(91 106)(92 107)(93 108)(94 109)(95 110)(96 111)(97 112)(98 113)(99 114)(100 115)(101 116)(102 117)(103 118)(104 119)(105 120)(121 144)(122 145)(123 146)(124 147)(125 148)(126 149)(127 150)(128 136)(129 137)(130 138)(131 139)(132 140)(133 141)(134 142)(135 143)(151 170)(152 171)(153 172)(154 173)(155 174)(156 175)(157 176)(158 177)(159 178)(160 179)(161 180)(162 166)(163 167)(164 168)(165 169)(181 203)(182 204)(183 205)(184 206)(185 207)(186 208)(187 209)(188 210)(189 196)(190 197)(191 198)(192 199)(193 200)(194 201)(195 202)(211 230)(212 231)(213 232)(214 233)(215 234)(216 235)(217 236)(218 237)(219 238)(220 239)(221 240)(222 226)(223 227)(224 228)(225 229)
(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)
(1 109)(2 108)(3 107)(4 106)(5 120)(6 119)(7 118)(8 117)(9 116)(10 115)(11 114)(12 113)(13 112)(14 111)(15 110)(16 95)(17 94)(18 93)(19 92)(20 91)(21 105)(22 104)(23 103)(24 102)(25 101)(26 100)(27 99)(28 98)(29 97)(30 96)(31 84)(32 83)(33 82)(34 81)(35 80)(36 79)(37 78)(38 77)(39 76)(40 90)(41 89)(42 88)(43 87)(44 86)(45 85)(46 65)(47 64)(48 63)(49 62)(50 61)(51 75)(52 74)(53 73)(54 72)(55 71)(56 70)(57 69)(58 68)(59 67)(60 66)(121 226)(122 240)(123 239)(124 238)(125 237)(126 236)(127 235)(128 234)(129 233)(130 232)(131 231)(132 230)(133 229)(134 228)(135 227)(136 215)(137 214)(138 213)(139 212)(140 211)(141 225)(142 224)(143 223)(144 222)(145 221)(146 220)(147 219)(148 218)(149 217)(150 216)(151 203)(152 202)(153 201)(154 200)(155 199)(156 198)(157 197)(158 196)(159 210)(160 209)(161 208)(162 207)(163 206)(164 205)(165 204)(166 185)(167 184)(168 183)(169 182)(170 181)(171 195)(172 194)(173 193)(174 192)(175 191)(176 190)(177 189)(178 188)(179 187)(180 186)

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

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

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

144 conjugacy classes

class 1 2A···2O2P···2AE 3 5A5B6A···6O10A···10AD15A15B15C15D30A···30BH
order12···22···23556···610···101515151530···30
size11···115···152222···22···222222···2

144 irreducible representations

dim111222222
type+++++++++
imageC1C2C2S3D5D6D10D15D30
kernelC24×D15C23×D15C23×C30C23×C10C23×C6C22×C10C22×C6C24C23
# reps1301121530460

Matrix representation of C24×D15 in GL6(𝔽31)

3000000
0300000
0030000
0003000
000010
000001
,
3000000
010000
0030000
0003000
0000300
0000030
,
100000
0300000
0030000
0003000
0000300
0000030
,
3000000
0300000
0030000
0003000
0000300
0000030
,
100000
010000
00212100
006900
00002311
00002015
,
3000000
0300000
00101000
00182100
0000514
00001626

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

׿
×
𝔽