Copied to
clipboard

?

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), (C23×C6)⋊7D5, C6⋊2(C23×D5), C10⋊2(S3×C23), (C23×C30)⋊5C2, (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

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

Generators and relations
 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 >

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

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

(1 43)(2 44)(3 45)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(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 98)(62 99)(63 100)(64 101)(65 102)(66 103)(67 104)(68 105)(69 91)(70 92)(71 93)(72 94)(73 95)(74 96)(75 97)(76 110)(77 111)(78 112)(79 113)(80 114)(81 115)(82 116)(83 117)(84 118)(85 119)(86 120)(87 106)(88 107)(89 108)(90 109)(121 152)(122 153)(123 154)(124 155)(125 156)(126 157)(127 158)(128 159)(129 160)(130 161)(131 162)(132 163)(133 164)(134 165)(135 151)(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)(181 218)(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)(196 237)(197 238)(198 239)(199 240)(200 226)(201 227)(202 228)(203 229)(204 230)(205 231)(206 232)(207 233)(208 234)(209 235)(210 236)

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

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

Matrix representation ►G ⊆ 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] >;
 

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

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
 

׿
×
⋊
ℤ
𝔽
○
≀