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G = C2×D6.D10order 480 = 25·3·5

Direct product of C2 and D6.D10

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

Aliases: C2×D6.D10, C30.11C24, C60.189C23, D30.35C23, Dic15.37C23, (C4×D5)⋊18D6, (C4×S3)⋊18D10, C62(C4○D20), C305(C4○D4), C15⋊Q818C22, C102(C4○D12), (C2×C20).343D6, C6.11(C23×D5), (S3×C20)⋊19C22, (C2×C12).347D10, (D5×C12)⋊19C22, (C4×D15)⋊25C22, C5⋊D1221C22, C3⋊D2020C22, C15⋊D421C22, C10.11(S3×C23), (C6×D5).38C23, (C22×D5).96D6, D6.24(C22×D5), (S3×C10).27C23, C20.186(C22×S3), (C2×C60).245C22, (C2×C30).230C23, (C2×Dic5).198D6, D10.40(C22×S3), (C22×S3).82D10, C12.186(C22×D5), (C2×Dic3).169D10, (C5×Dic3).28C23, Dic3.25(C22×D5), Dic5.42(C22×S3), (C3×Dic5).40C23, (C6×Dic5).227C22, (C10×Dic3).209C22, (C2×Dic15).234C22, (C22×D15).119C22, (S3×C2×C20)⋊9C2, (D5×C2×C12)⋊9C2, (C2×C4×D5)⋊15S3, (S3×C2×C4)⋊15D5, C32(C2×C4○D20), C52(C2×C4○D12), C155(C2×C4○D4), (C2×C4×D15)⋊30C2, (C2×C15⋊Q8)⋊26C2, C4.159(C2×S3×D5), (C2×C3⋊D20)⋊24C2, (C2×C15⋊D4)⋊24C2, (C2×C5⋊D12)⋊24C2, C2.15(C22×S3×D5), C22.99(C2×S3×D5), (C2×C4).248(S3×D5), (D5×C2×C6).115C22, (S3×C2×C10).100C22, (C2×C6).240(C22×D5), (C2×C10).240(C22×S3), SmallGroup(480,1083)

Series: Derived Chief Lower central Upper central

C1C30 — C2×D6.D10
C1C5C15C30C6×D5C15⋊D4C2×C15⋊D4 — C2×D6.D10
C15C30 — C2×D6.D10
C1C2×C4

Generators and relations for C2×D6.D10
 G = < a,b,c,d,e | a2=b6=c2=1, d10=e2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=d9 >

Subgroups: 1564 in 328 conjugacy classes, 116 normal (36 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×2], C4 [×6], C22, C22 [×12], C5, S3 [×4], C6, C6 [×2], C6 [×2], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], D5 [×4], C10, C10 [×2], C10 [×2], Dic3 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×2], D6 [×6], C2×C6, C2×C6 [×4], C15, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×6], C2×C10, C2×C10 [×4], Dic6 [×4], C4×S3 [×4], C4×S3 [×4], D12 [×4], C2×Dic3, C2×Dic3, C3⋊D4 [×8], C2×C12, C2×C12 [×5], C22×S3, C22×S3, C22×C6, C5×S3 [×2], C3×D5 [×2], D15 [×2], C30, C30 [×2], C2×C4○D4, Dic10 [×4], C4×D5 [×4], C4×D5 [×4], D20 [×4], C2×Dic5, C2×Dic5, C5⋊D4 [×8], C2×C20, C2×C20 [×5], C22×D5, C22×D5, C22×C10, C2×Dic6, S3×C2×C4, S3×C2×C4, C2×D12, C4○D12 [×8], C2×C3⋊D4 [×2], C22×C12, C5×Dic3 [×2], C3×Dic5 [×2], Dic15 [×2], C60 [×2], C6×D5 [×2], C6×D5 [×2], S3×C10 [×2], S3×C10 [×2], D30 [×2], D30 [×2], C2×C30, C2×Dic10, C2×C4×D5, C2×C4×D5, C2×D20, C4○D20 [×8], C2×C5⋊D4 [×2], C22×C20, C2×C4○D12, C15⋊D4 [×4], C3⋊D20 [×4], C5⋊D12 [×4], C15⋊Q8 [×4], D5×C12 [×4], C6×Dic5, S3×C20 [×4], C10×Dic3, C4×D15 [×4], C2×Dic15, C2×C60, D5×C2×C6, S3×C2×C10, C22×D15, C2×C4○D20, D6.D10 [×8], C2×C15⋊D4, C2×C3⋊D20, C2×C5⋊D12, C2×C15⋊Q8, D5×C2×C12, S3×C2×C20, C2×C4×D15, C2×D6.D10
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C4○D4 [×2], C24, D10 [×7], C22×S3 [×7], C2×C4○D4, C22×D5 [×7], C4○D12 [×2], S3×C23, S3×D5, C4○D20 [×2], C23×D5, C2×C4○D12, C2×S3×D5 [×3], C2×C4○D20, D6.D10 [×2], C22×S3×D5, C2×D6.D10

Smallest permutation representation of C2×D6.D10
On 240 points
Generators in S240
(1 186)(2 187)(3 188)(4 189)(5 190)(6 191)(7 192)(8 193)(9 194)(10 195)(11 196)(12 197)(13 198)(14 199)(15 200)(16 181)(17 182)(18 183)(19 184)(20 185)(21 110)(22 111)(23 112)(24 113)(25 114)(26 115)(27 116)(28 117)(29 118)(30 119)(31 120)(32 101)(33 102)(34 103)(35 104)(36 105)(37 106)(38 107)(39 108)(40 109)(41 238)(42 239)(43 240)(44 221)(45 222)(46 223)(47 224)(48 225)(49 226)(50 227)(51 228)(52 229)(53 230)(54 231)(55 232)(56 233)(57 234)(58 235)(59 236)(60 237)(61 210)(62 211)(63 212)(64 213)(65 214)(66 215)(67 216)(68 217)(69 218)(70 219)(71 220)(72 201)(73 202)(74 203)(75 204)(76 205)(77 206)(78 207)(79 208)(80 209)(81 141)(82 142)(83 143)(84 144)(85 145)(86 146)(87 147)(88 148)(89 149)(90 150)(91 151)(92 152)(93 153)(94 154)(95 155)(96 156)(97 157)(98 158)(99 159)(100 160)(121 169)(122 170)(123 171)(124 172)(125 173)(126 174)(127 175)(128 176)(129 177)(130 178)(131 179)(132 180)(133 161)(134 162)(135 163)(136 164)(137 165)(138 166)(139 167)(140 168)
(1 204 48 11 214 58)(2 205 49 12 215 59)(3 206 50 13 216 60)(4 207 51 14 217 41)(5 208 52 15 218 42)(6 209 53 16 219 43)(7 210 54 17 220 44)(8 211 55 18 201 45)(9 212 56 19 202 46)(10 213 57 20 203 47)(21 140 155 31 130 145)(22 121 156 32 131 146)(23 122 157 33 132 147)(24 123 158 34 133 148)(25 124 159 35 134 149)(26 125 160 36 135 150)(27 126 141 37 136 151)(28 127 142 38 137 152)(29 128 143 39 138 153)(30 129 144 40 139 154)(61 231 182 71 221 192)(62 232 183 72 222 193)(63 233 184 73 223 194)(64 234 185 74 224 195)(65 235 186 75 225 196)(66 236 187 76 226 197)(67 237 188 77 227 198)(68 238 189 78 228 199)(69 239 190 79 229 200)(70 240 191 80 230 181)(81 106 164 91 116 174)(82 107 165 92 117 175)(83 108 166 93 118 176)(84 109 167 94 119 177)(85 110 168 95 120 178)(86 111 169 96 101 179)(87 112 170 97 102 180)(88 113 171 98 103 161)(89 114 172 99 104 162)(90 115 173 100 105 163)
(1 158)(2 159)(3 160)(4 141)(5 142)(6 143)(7 144)(8 145)(9 146)(10 147)(11 148)(12 149)(13 150)(14 151)(15 152)(16 153)(17 154)(18 155)(19 156)(20 157)(21 45)(22 46)(23 47)(24 48)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 41)(38 42)(39 43)(40 44)(61 177)(62 178)(63 179)(64 180)(65 161)(66 162)(67 163)(68 164)(69 165)(70 166)(71 167)(72 168)(73 169)(74 170)(75 171)(76 172)(77 173)(78 174)(79 175)(80 176)(81 189)(82 190)(83 191)(84 192)(85 193)(86 194)(87 195)(88 196)(89 197)(90 198)(91 199)(92 200)(93 181)(94 182)(95 183)(96 184)(97 185)(98 186)(99 187)(100 188)(101 233)(102 234)(103 235)(104 236)(105 237)(106 238)(107 239)(108 240)(109 221)(110 222)(111 223)(112 224)(113 225)(114 226)(115 227)(116 228)(117 229)(118 230)(119 231)(120 232)(121 202)(122 203)(123 204)(124 205)(125 206)(126 207)(127 208)(128 209)(129 210)(130 211)(131 212)(132 213)(133 214)(134 215)(135 216)(136 217)(137 218)(138 219)(139 220)(140 201)
(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 10 11 20)(2 19 12 9)(3 8 13 18)(4 17 14 7)(5 6 15 16)(21 36 31 26)(22 25 32 35)(23 34 33 24)(27 30 37 40)(28 39 38 29)(41 54 51 44)(42 43 52 53)(45 50 55 60)(46 59 56 49)(47 48 57 58)(61 78 71 68)(62 67 72 77)(63 76 73 66)(64 65 74 75)(69 70 79 80)(81 84 91 94)(82 93 92 83)(85 100 95 90)(86 89 96 99)(87 98 97 88)(101 104 111 114)(102 113 112 103)(105 120 115 110)(106 109 116 119)(107 118 117 108)(121 124 131 134)(122 133 132 123)(125 140 135 130)(126 129 136 139)(127 138 137 128)(141 144 151 154)(142 153 152 143)(145 160 155 150)(146 149 156 159)(147 158 157 148)(161 180 171 170)(162 169 172 179)(163 178 173 168)(164 167 174 177)(165 176 175 166)(181 190 191 200)(182 199 192 189)(183 188 193 198)(184 197 194 187)(185 186 195 196)(201 206 211 216)(202 215 212 205)(203 204 213 214)(207 220 217 210)(208 209 218 219)(221 238 231 228)(222 227 232 237)(223 236 233 226)(224 225 234 235)(229 230 239 240)

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

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

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

84 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D4E4F4G4H4I4J5A5B6A6B6C6D6E6F6G10A···10F10G···10N12A12B12C12D12E12F12G12H15A15B20A···20H20I···20P30A···30F60A···60H
order12222222223444444444455666666610···1010···101212121212121212151520···2020···2030···3060···60
size1111661010303021111661010303022222101010102···26···6222210101010442···26···64···44···4

84 irreducible representations

dim11111111122222222222224444
type++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2S3D5D6D6D6D6C4○D4D10D10D10D10C4○D12C4○D20S3×D5C2×S3×D5C2×S3×D5D6.D10
kernelC2×D6.D10D6.D10C2×C15⋊D4C2×C3⋊D20C2×C5⋊D12C2×C15⋊Q8D5×C2×C12S3×C2×C20C2×C4×D15C2×C4×D5S3×C2×C4C4×D5C2×Dic5C2×C20C22×D5C30C4×S3C2×Dic3C2×C12C22×S3C10C6C2×C4C4C22C2
# reps181111111124111482228162428

Matrix representation of C2×D6.D10 in GL5(𝔽61)

600000
01000
00100
00010
00001
,
10000
01000
00100
0004846
000014
,
10000
01000
00100
0003639
0004525
,
600000
014400
0171700
000500
000050
,
10000
014400
006000
0005042
000011

G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,48,0,0,0,0,46,14],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,36,45,0,0,0,39,25],[60,0,0,0,0,0,1,17,0,0,0,44,17,0,0,0,0,0,50,0,0,0,0,0,50],[1,0,0,0,0,0,1,0,0,0,0,44,60,0,0,0,0,0,50,0,0,0,0,42,11] >;

C2×D6.D10 in GAP, Magma, Sage, TeX

C_2\times D_6.D_{10}
% in TeX

G:=Group("C2xD6.D10");
// GroupNames label

G:=SmallGroup(480,1083);
// by ID

G=gap.SmallGroup(480,1083);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^6=c^2=1,d^10=e^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=b^3*c,e*d*e^-1=d^9>;
// generators/relations

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