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G = C22×C4×D15  order 480 = 25·3·5

Direct product of C22×C4 and D15

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

Aliases: C22×C4×D15, C60⋊10C23, C30.55C24, C23.39D30, D30.44C23, Dic15⋊11C23, (C2×C20)⋊35D6, C15⋊8(C23×C4), (C2×C12)⋊35D10, C20⋊9(C22×S3), C30⋊8(C22×C4), C12⋊9(C22×D5), (C22×C60)⋊14C2, (C22×C20)⋊14S3, (C2×C60)⋊46C22, (C22×C12)⋊10D5, C6.55(C23×D5), C2.1(C23×D15), C10.55(S3×C23), (C23×D15).6C2, (C2×C30).319C23, (C22×C6).125D10, (C22×C10).143D6, (C2×Dic15)⋊39C22, (C22×Dic15)⋊21C2, C22.29(C22×D15), (C22×C30).148C22, (C22×D15).127C22, C6⋊3(C2×C4×D5), C10⋊4(S3×C2×C4), C5⋊4(S3×C22×C4), C3⋊3(D5×C22×C4), (C2×C6)⋊12(C4×D5), (C2×C30)⋊34(C2×C4), (C2×C10)⋊21(C4×S3), (C2×C6).315(C22×D5), (C2×C10).314(C22×S3), SmallGroup(480,1166)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C15 — C22×C4×D15
Chief series
C1 — C5 — C15 — C30 — D30 — C22×D15 — C23×D15 — C22×C4×D15
Lower central
C15 — C22×C4×D15

Subgroups: 2292 in 472 conjugacy classes, 199 normal (19 characteristic)
C1, C2, C2 [×6], C2 [×8], C3, C4 [×4], C4 [×4], C22 [×7], C22 [×28], C5, S3 [×8], C6, C6 [×6], C2×C4 [×6], C2×C4 [×22], C23, C23 [×14], D5 [×8], C10, C10 [×6], Dic3 [×4], C12 [×4], D6 [×28], C2×C6 [×7], C15, C22×C4, C22×C4 [×13], C24, Dic5 [×4], C20 [×4], D10 [×28], C2×C10 [×7], C4×S3 [×16], C2×Dic3 [×6], C2×C12 [×6], C22×S3 [×14], C22×C6, D15 [×8], C30, C30 [×6], C23×C4, C4×D5 [×16], C2×Dic5 [×6], C2×C20 [×6], C22×D5 [×14], C22×C10, S3×C2×C4 [×12], C22×Dic3, C22×C12, S3×C23, Dic15 [×4], C60 [×4], D30 [×28], C2×C30 [×7], C2×C4×D5 [×12], C22×Dic5, C22×C20, C23×D5, S3×C22×C4, C4×D15 [×16], C2×Dic15 [×6], C2×C60 [×6], C22×D15 [×14], C22×C30, D5×C22×C4, C2×C4×D15 [×12], C22×Dic15, C22×C60, C23×D15, C22×C4×D15

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], C23 [×15], D5, D6 [×7], C22×C4 [×14], C24, D10 [×7], C4×S3 [×4], C22×S3 [×7], D15, C23×C4, C4×D5 [×4], C22×D5 [×7], S3×C2×C4 [×6], S3×C23, D30 [×7], C2×C4×D5 [×6], C23×D5, S3×C22×C4, C4×D15 [×4], C22×D15 [×7], D5×C22×C4, C2×C4×D15 [×6], C23×D15, C22×C4×D15

Generators and relations
 G = < a,b,c,d,e | a2=b2=c4=d15=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

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

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

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

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

Matrix representation ►G ⊆ GL4(𝔽61) generated by

1000
06000
0010
0001
,
60000
06000
00600
00060
,
11000
0100
0010
0001
,
1000
0100
002314
003945
,
60000
0100
00160
00060
G:=sub<GL(4,GF(61))| [1,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[11,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,23,39,0,0,14,45],[60,0,0,0,0,1,0,0,0,0,1,0,0,0,60,60] >;
 

144 conjugacy classes

class 1 2A···2G2H···2O 3 4A···4H4I···4P5A5B6A···6G10A···10N12A···12H15A15B15C15D20A···20P30A···30AB60A···60AF
order12···22···234···44···4556···610···1012···121515151520···2030···3060···60
size11···115···1521···115···15222···22···22···222222···22···22···2

144 irreducible representations

dim111111222222222222
type++++++++++++++
imageC1C2C2C2C2C4S3D5D6D6D10D10C4×S3D15C4×D5D30D30C4×D15
kernelC22×C4×D15C2×C4×D15C22×Dic15C22×C60C23×D15C22×D15C22×C20C22×C12C2×C20C22×C10C2×C12C22×C6C2×C10C22×C4C2×C6C2×C4C23C22
# reps112111161261122841624432

In GAP, Magma, Sage, TeX 

C_2^2\times C_4\times D_{15}
 
% in TeX
 
G:=Group("C2^2xC4xD15");
 
// GroupNames label
 
G:=SmallGroup(480,1166);
 
// by ID
 
G=gap.SmallGroup(480,1166);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,80,2693,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^4=d^15=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
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