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G = D5×C22×C12  order 480 = 25·3·5

Direct product of C22×C12 and D5

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

Aliases: D5×C22×C12, C60⋊13C23, C30.70C24, C5⋊2(C23×C12), C15⋊9(C23×C4), C30⋊9(C22×C4), C20⋊3(C22×C6), (C2×C60)⋊51C22, (C22×C20)⋊14C6, (C22×C60)⋊22C2, C10⋊2(C22×C12), C10.2(C23×C6), (C23×D5).7C6, C6.70(C23×D5), C23.39(C6×D5), Dic5⋊3(C22×C6), (C6×D5).74C23, (C2×C30).380C23, (C22×Dic5)⋊13C6, (C6×Dic5)⋊39C22, (C3×Dic5)⋊11C23, D10.15(C22×C6), (C22×C6).136D10, (C22×C30).165C22, (C2×C20)⋊14(C2×C6), (C2×C30)⋊36(C2×C4), C2.1(D5×C22×C6), (C2×C6×Dic5)⋊21C2, (C2×C10)⋊13(C2×C12), C22.29(D5×C2×C6), (D5×C22×C6).10C2, (C2×Dic5)⋊12(C2×C6), (D5×C2×C6).156C22, (C22×C10).52(C2×C6), (C2×C10).63(C22×C6), (C22×D5).45(C2×C6), (C2×C6).376(C22×D5), SmallGroup(480,1136)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C5 — D5×C22×C12
Chief series
C1 — C5 — C10 — C30 — C6×D5 — D5×C2×C6 — D5×C22×C6 — D5×C22×C12
Lower central
C5 — D5×C22×C12

Subgroups: 1200 in 472 conjugacy classes, 290 normal (22 characteristic)
C1, C2, C2 [×6], C2 [×8], C3, C4 [×4], C4 [×4], C22 [×7], C22 [×28], C5, C6, C6 [×6], C6 [×8], C2×C4 [×6], C2×C4 [×22], C23, C23 [×14], D5 [×8], C10, C10 [×6], C12 [×4], C12 [×4], C2×C6 [×7], C2×C6 [×28], C15, C22×C4, C22×C4 [×13], C24, Dic5 [×4], C20 [×4], D10 [×28], C2×C10 [×7], C2×C12 [×6], C2×C12 [×22], C22×C6, C22×C6 [×14], C3×D5 [×8], C30, C30 [×6], C23×C4, C4×D5 [×16], C2×Dic5 [×6], C2×C20 [×6], C22×D5 [×14], C22×C10, C22×C12, C22×C12 [×13], C23×C6, C3×Dic5 [×4], C60 [×4], C6×D5 [×28], C2×C30 [×7], C2×C4×D5 [×12], C22×Dic5, C22×C20, C23×D5, C23×C12, D5×C12 [×16], C6×Dic5 [×6], C2×C60 [×6], D5×C2×C6 [×14], C22×C30, D5×C22×C4, D5×C2×C12 [×12], C2×C6×Dic5, C22×C60, D5×C22×C6, D5×C22×C12

Quotients:
C1, C2 [×15], C3, C4 [×8], C22 [×35], C6 [×15], C2×C4 [×28], C23 [×15], D5, C12 [×8], C2×C6 [×35], C22×C4 [×14], C24, D10 [×7], C2×C12 [×28], C22×C6 [×15], C3×D5, C23×C4, C4×D5 [×4], C22×D5 [×7], C22×C12 [×14], C23×C6, C6×D5 [×7], C2×C4×D5 [×6], C23×D5, C23×C12, D5×C12 [×4], D5×C2×C6 [×7], D5×C22×C4, D5×C2×C12 [×6], D5×C22×C6, D5×C22×C12

Generators and relations
 G = < a,b,c,d,e | a2=b2=c12=d5=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
(1 137)(2 138)(3 139)(4 140)(5 141)(6 142)(7 143)(8 144)(9 133)(10 134)(11 135)(12 136)(13 200)(14 201)(15 202)(16 203)(17 204)(18 193)(19 194)(20 195)(21 196)(22 197)(23 198)(24 199)(25 153)(26 154)(27 155)(28 156)(29 145)(30 146)(31 147)(32 148)(33 149)(34 150)(35 151)(36 152)(37 121)(38 122)(39 123)(40 124)(41 125)(42 126)(43 127)(44 128)(45 129)(46 130)(47 131)(48 132)(49 157)(50 158)(51 159)(52 160)(53 161)(54 162)(55 163)(56 164)(57 165)(58 166)(59 167)(60 168)(61 227)(62 228)(63 217)(64 218)(65 219)(66 220)(67 221)(68 222)(69 223)(70 224)(71 225)(72 226)(73 216)(74 205)(75 206)(76 207)(77 208)(78 209)(79 210)(80 211)(81 212)(82 213)(83 214)(84 215)(85 236)(86 237)(87 238)(88 239)(89 240)(90 229)(91 230)(92 231)(93 232)(94 233)(95 234)(96 235)(97 188)(98 189)(99 190)(100 191)(101 192)(102 181)(103 182)(104 183)(105 184)(106 185)(107 186)(108 187)(109 174)(110 175)(111 176)(112 177)(113 178)(114 179)(115 180)(116 169)(117 170)(118 171)(119 172)(120 173)

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

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

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

Matrix representation ►G ⊆ GL4(𝔽61) generated by

60000
06000
00600
00060
,
60000
0100
00600
00060
,
1000
06000
00210
00021
,
1000
0100
0001
006043
,
60000
0100
0001
0010
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,0,60,0,0,0,0,21,0,0,0,0,21],[1,0,0,0,0,1,0,0,0,0,0,60,0,0,1,43],[60,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0] >;
 

192 conjugacy classes

class 1 2A···2G2H···2O3A3B4A···4H4I···4P5A5B6A···6N6O···6AD10A···10N12A···12P12Q···12AF15A15B15C15D20A···20P30A···30AB60A···60AF
order12···22···2334···44···4556···66···610···1012···1212···121515151520···2030···3060···60
size11···15···5111···15···5221···15···52···21···15···522222···22···22···2

192 irreducible representations

dim11111111111122222222
type++++++++
imageC1C2C2C2C2C3C4C6C6C6C6C12D5D10D10C3×D5C4×D5C6×D5C6×D5D5×C12
kernelD5×C22×C12D5×C2×C12C2×C6×Dic5C22×C60D5×C22×C6D5×C22×C4D5×C2×C6C2×C4×D5C22×Dic5C22×C20C23×D5C22×D5C22×C12C2×C12C22×C6C22×C4C2×C6C2×C4C23C22
# reps1121112162422232212241624432

In GAP, Magma, Sage, TeX 

D_5\times C_2^2\times C_{12}
 
% in TeX
 
G:=Group("D5xC2^2xC12");
 
// GroupNames label
 
G:=SmallGroup(480,1136);
 
// by ID
 
G=gap.SmallGroup(480,1136);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-5,192,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^12=d^5=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
 

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