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G = D5×C22×C12order 480 = 25·3·5

Direct product of C22×C12 and D5

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

Aliases: D5×C22×C12, C6013C23, C30.70C24, C52(C23×C12), C159(C23×C4), C309(C22×C4), C203(C22×C6), (C22×C60)⋊22C2, C102(C22×C12), (C22×C20)⋊14C6, (C2×C60)⋊51C22, C10.2(C23×C6), (C23×D5).7C6, C6.70(C23×D5), C23.39(C6×D5), Dic53(C22×C6), (C6×D5).74C23, (C2×C30).380C23, (C3×Dic5)⋊11C23, (C6×Dic5)⋊39C22, (C22×Dic5)⋊13C6, D10.15(C22×C6), (C22×C6).136D10, (C22×C30).165C22, (C2×C30)⋊36(C2×C4), (C2×C20)⋊14(C2×C6), 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, (C2×C10).63(C22×C6), (C22×C10).52(C2×C6), (C22×D5).45(C2×C6), (C2×C6).376(C22×D5), SmallGroup(480,1136)

Series: Derived Chief Lower central Upper central

C1C5 — D5×C22×C12
C1C5C10C30C6×D5D5×C2×C6D5×C22×C6 — D5×C22×C12
C5 — D5×C22×C12
C1C22×C12

Generators and relations for D5×C22×C12
 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 >

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

Smallest permutation representation of D5×C22×C12
On 240 points
Generators in S240
(1 65)(2 66)(3 67)(4 68)(5 69)(6 70)(7 71)(8 72)(9 61)(10 62)(11 63)(12 64)(13 101)(14 102)(15 103)(16 104)(17 105)(18 106)(19 107)(20 108)(21 97)(22 98)(23 99)(24 100)(25 132)(26 121)(27 122)(28 123)(29 124)(30 125)(31 126)(32 127)(33 128)(34 129)(35 130)(36 131)(37 159)(38 160)(39 161)(40 162)(41 163)(42 164)(43 165)(44 166)(45 167)(46 168)(47 157)(48 158)(49 184)(50 185)(51 186)(52 187)(53 188)(54 189)(55 190)(56 191)(57 192)(58 181)(59 182)(60 183)(73 221)(74 222)(75 223)(76 224)(77 225)(78 226)(79 227)(80 228)(81 217)(82 218)(83 219)(84 220)(85 171)(86 172)(87 173)(88 174)(89 175)(90 176)(91 177)(92 178)(93 179)(94 180)(95 169)(96 170)(109 153)(110 154)(111 155)(112 156)(113 145)(114 146)(115 147)(116 148)(117 149)(118 150)(119 151)(120 152)(133 229)(134 230)(135 231)(136 232)(137 233)(138 234)(139 235)(140 236)(141 237)(142 238)(143 239)(144 240)(193 216)(194 205)(195 206)(196 207)(197 208)(198 209)(199 210)(200 211)(201 212)(202 213)(203 214)(204 215)
(1 237)(2 238)(3 239)(4 240)(5 229)(6 230)(7 231)(8 232)(9 233)(10 234)(11 235)(12 236)(13 171)(14 172)(15 173)(16 174)(17 175)(18 176)(19 177)(20 178)(21 179)(22 180)(23 169)(24 170)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 49)(35 50)(36 51)(37 202)(38 203)(39 204)(40 193)(41 194)(42 195)(43 196)(44 197)(45 198)(46 199)(47 200)(48 201)(61 137)(62 138)(63 139)(64 140)(65 141)(66 142)(67 143)(68 144)(69 133)(70 134)(71 135)(72 136)(73 110)(74 111)(75 112)(76 113)(77 114)(78 115)(79 116)(80 117)(81 118)(82 119)(83 120)(84 109)(85 101)(86 102)(87 103)(88 104)(89 105)(90 106)(91 107)(92 108)(93 97)(94 98)(95 99)(96 100)(121 188)(122 189)(123 190)(124 191)(125 192)(126 181)(127 182)(128 183)(129 184)(130 185)(131 186)(132 187)(145 224)(146 225)(147 226)(148 227)(149 228)(150 217)(151 218)(152 219)(153 220)(154 221)(155 222)(156 223)(157 211)(158 212)(159 213)(160 214)(161 215)(162 216)(163 205)(164 206)(165 207)(166 208)(167 209)(168 210)
(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 23 166 223 55)(2 24 167 224 56)(3 13 168 225 57)(4 14 157 226 58)(5 15 158 227 59)(6 16 159 228 60)(7 17 160 217 49)(8 18 161 218 50)(9 19 162 219 51)(10 20 163 220 52)(11 21 164 221 53)(12 22 165 222 54)(25 234 178 205 153)(26 235 179 206 154)(27 236 180 207 155)(28 237 169 208 156)(29 238 170 209 145)(30 239 171 210 146)(31 240 172 211 147)(32 229 173 212 148)(33 230 174 213 149)(34 231 175 214 150)(35 232 176 215 151)(36 233 177 216 152)(37 80 183 70 104)(38 81 184 71 105)(39 82 185 72 106)(40 83 186 61 107)(41 84 187 62 108)(42 73 188 63 97)(43 74 189 64 98)(44 75 190 65 99)(45 76 191 66 100)(46 77 192 67 101)(47 78 181 68 102)(48 79 182 69 103)(85 199 114 125 143)(86 200 115 126 144)(87 201 116 127 133)(88 202 117 128 134)(89 203 118 129 135)(90 204 119 130 136)(91 193 120 131 137)(92 194 109 132 138)(93 195 110 121 139)(94 196 111 122 140)(95 197 112 123 141)(96 198 113 124 142)
(1 123)(2 124)(3 125)(4 126)(5 127)(6 128)(7 129)(8 130)(9 131)(10 132)(11 121)(12 122)(13 114)(14 115)(15 116)(16 117)(17 118)(18 119)(19 120)(20 109)(21 110)(22 111)(23 112)(24 113)(25 62)(26 63)(27 64)(28 65)(29 66)(30 67)(31 68)(32 69)(33 70)(34 71)(35 72)(36 61)(37 213)(38 214)(39 215)(40 216)(41 205)(42 206)(43 207)(44 208)(45 209)(46 210)(47 211)(48 212)(49 135)(50 136)(51 137)(52 138)(53 139)(54 140)(55 141)(56 142)(57 143)(58 144)(59 133)(60 134)(73 179)(74 180)(75 169)(76 170)(77 171)(78 172)(79 173)(80 174)(81 175)(82 176)(83 177)(84 178)(85 225)(86 226)(87 227)(88 228)(89 217)(90 218)(91 219)(92 220)(93 221)(94 222)(95 223)(96 224)(97 154)(98 155)(99 156)(100 145)(101 146)(102 147)(103 148)(104 149)(105 150)(106 151)(107 152)(108 153)(157 200)(158 201)(159 202)(160 203)(161 204)(162 193)(163 194)(164 195)(165 196)(166 197)(167 198)(168 199)(181 240)(182 229)(183 230)(184 231)(185 232)(186 233)(187 234)(188 235)(189 236)(190 237)(191 238)(192 239)

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

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

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

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

Matrix representation of D5×C22×C12 in 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] >;

D5×C22×C12 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

׿
×
𝔽