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## G = C2×S3×Dic10order 480 = 25·3·5

### Direct product of C2, S3 and Dic10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C2×S3×Dic10
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — S3×Dic5 — C2×S3×Dic5 — C2×S3×Dic10
 Lower central C15 — C30 — C2×S3×Dic10
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×S3×Dic10
G = < a,b,c,d,e | a2=b3=c2=d20=1, e2=d10, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 1276 in 312 conjugacy classes, 132 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C4 [×10], C22, C22 [×6], C5, S3 [×4], C6, C6 [×2], C2×C4, C2×C4 [×17], Q8 [×16], C23, C10, C10 [×2], C10 [×4], Dic3 [×2], Dic3 [×4], C12 [×2], C12 [×4], D6 [×6], C2×C6, C15, C22×C4 [×3], C2×Q8 [×12], Dic5 [×4], Dic5 [×4], C20 [×2], C20 [×2], C2×C10, C2×C10 [×6], Dic6 [×12], C4×S3 [×4], C4×S3 [×8], C2×Dic3, C2×Dic3 [×2], C2×C12, C2×C12 [×2], C3×Q8 [×4], C22×S3, C5×S3 [×4], C30, C30 [×2], C22×Q8, Dic10 [×4], Dic10 [×12], C2×Dic5 [×2], C2×Dic5 [×10], C2×C20, C2×C20 [×5], C22×C10, C2×Dic6 [×3], S3×C2×C4, S3×C2×C4 [×2], S3×Q8 [×8], C6×Q8, C5×Dic3 [×2], C3×Dic5 [×4], Dic15 [×4], C60 [×2], S3×C10 [×6], C2×C30, C2×Dic10, C2×Dic10 [×11], C22×Dic5 [×2], C22×C20, C2×S3×Q8, S3×Dic5 [×8], C15⋊Q8 [×8], C3×Dic10 [×4], C6×Dic5 [×2], S3×C20 [×4], C10×Dic3, Dic30 [×4], C2×Dic15 [×2], C2×C60, S3×C2×C10, C22×Dic10, S3×Dic10 [×8], C2×S3×Dic5 [×2], C2×C15⋊Q8 [×2], C6×Dic10, S3×C2×C20, C2×Dic30, C2×S3×Dic10
Quotients: C1, C2 [×15], C22 [×35], S3, Q8 [×4], C23 [×15], D5, D6 [×7], C2×Q8 [×6], C24, D10 [×7], C22×S3 [×7], C22×Q8, Dic10 [×4], C22×D5 [×7], S3×Q8 [×2], S3×C23, S3×D5, C2×Dic10 [×6], C23×D5, C2×S3×Q8, C2×S3×D5 [×3], C22×Dic10, S3×Dic10 [×2], C22×S3×D5, C2×S3×Dic10

Smallest permutation representation of C2×S3×Dic10
On 240 points
Generators in S240
(1 55)(2 56)(3 57)(4 58)(5 59)(6 60)(7 41)(8 42)(9 43)(10 44)(11 45)(12 46)(13 47)(14 48)(15 49)(16 50)(17 51)(18 52)(19 53)(20 54)(21 107)(22 108)(23 109)(24 110)(25 111)(26 112)(27 113)(28 114)(29 115)(30 116)(31 117)(32 118)(33 119)(34 120)(35 101)(36 102)(37 103)(38 104)(39 105)(40 106)(61 190)(62 191)(63 192)(64 193)(65 194)(66 195)(67 196)(68 197)(69 198)(70 199)(71 200)(72 181)(73 182)(74 183)(75 184)(76 185)(77 186)(78 187)(79 188)(80 189)(81 214)(82 215)(83 216)(84 217)(85 218)(86 219)(87 220)(88 201)(89 202)(90 203)(91 204)(92 205)(93 206)(94 207)(95 208)(96 209)(97 210)(98 211)(99 212)(100 213)(121 228)(122 229)(123 230)(124 231)(125 232)(126 233)(127 234)(128 235)(129 236)(130 237)(131 238)(132 239)(133 240)(134 221)(135 222)(136 223)(137 224)(138 225)(139 226)(140 227)(141 163)(142 164)(143 165)(144 166)(145 167)(146 168)(147 169)(148 170)(149 171)(150 172)(151 173)(152 174)(153 175)(154 176)(155 177)(156 178)(157 179)(158 180)(159 161)(160 162)
(1 109 163)(2 110 164)(3 111 165)(4 112 166)(5 113 167)(6 114 168)(7 115 169)(8 116 170)(9 117 171)(10 118 172)(11 119 173)(12 120 174)(13 101 175)(14 102 176)(15 103 177)(16 104 178)(17 105 179)(18 106 180)(19 107 161)(20 108 162)(21 159 53)(22 160 54)(23 141 55)(24 142 56)(25 143 57)(26 144 58)(27 145 59)(28 146 60)(29 147 41)(30 148 42)(31 149 43)(32 150 44)(33 151 45)(34 152 46)(35 153 47)(36 154 48)(37 155 49)(38 156 50)(39 157 51)(40 158 52)(61 240 87)(62 221 88)(63 222 89)(64 223 90)(65 224 91)(66 225 92)(67 226 93)(68 227 94)(69 228 95)(70 229 96)(71 230 97)(72 231 98)(73 232 99)(74 233 100)(75 234 81)(76 235 82)(77 236 83)(78 237 84)(79 238 85)(80 239 86)(121 208 198)(122 209 199)(123 210 200)(124 211 181)(125 212 182)(126 213 183)(127 214 184)(128 215 185)(129 216 186)(130 217 187)(131 218 188)(132 219 189)(133 220 190)(134 201 191)(135 202 192)(136 203 193)(137 204 194)(138 205 195)(139 206 196)(140 207 197)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 149)(22 150)(23 151)(24 152)(25 153)(26 154)(27 155)(28 156)(29 157)(30 158)(31 159)(32 160)(33 141)(34 142)(35 143)(36 144)(37 145)(38 146)(39 147)(40 148)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(81 224)(82 225)(83 226)(84 227)(85 228)(86 229)(87 230)(88 231)(89 232)(90 233)(91 234)(92 235)(93 236)(94 237)(95 238)(96 239)(97 240)(98 221)(99 222)(100 223)(101 165)(102 166)(103 167)(104 168)(105 169)(106 170)(107 171)(108 172)(109 173)(110 174)(111 175)(112 176)(113 177)(114 178)(115 179)(116 180)(117 161)(118 162)(119 163)(120 164)(121 218)(122 219)(123 220)(124 201)(125 202)(126 203)(127 204)(128 205)(129 206)(130 207)(131 208)(132 209)(133 210)(134 211)(135 212)(136 213)(137 214)(138 215)(139 216)(140 217)(181 191)(182 192)(183 193)(184 194)(185 195)(186 196)(187 197)(188 198)(189 199)(190 200)
(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)
(1 199 11 189)(2 198 12 188)(3 197 13 187)(4 196 14 186)(5 195 15 185)(6 194 16 184)(7 193 17 183)(8 192 18 182)(9 191 19 181)(10 190 20 200)(21 231 31 221)(22 230 32 240)(23 229 33 239)(24 228 34 238)(25 227 35 237)(26 226 36 236)(27 225 37 235)(28 224 38 234)(29 223 39 233)(30 222 40 232)(41 64 51 74)(42 63 52 73)(43 62 53 72)(44 61 54 71)(45 80 55 70)(46 79 56 69)(47 78 57 68)(48 77 58 67)(49 76 59 66)(50 75 60 65)(81 146 91 156)(82 145 92 155)(83 144 93 154)(84 143 94 153)(85 142 95 152)(86 141 96 151)(87 160 97 150)(88 159 98 149)(89 158 99 148)(90 157 100 147)(101 130 111 140)(102 129 112 139)(103 128 113 138)(104 127 114 137)(105 126 115 136)(106 125 116 135)(107 124 117 134)(108 123 118 133)(109 122 119 132)(110 121 120 131)(161 211 171 201)(162 210 172 220)(163 209 173 219)(164 208 174 218)(165 207 175 217)(166 206 176 216)(167 205 177 215)(168 204 178 214)(169 203 179 213)(170 202 180 212)

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

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

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

78 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 5A 5B 6A 6B 6C 10A ··· 10F 10G ··· 10N 12A 12B 12C 12D 12E 12F 15A 15B 20A ··· 20H 20I ··· 20P 30A ··· 30F 60A ··· 60H order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 5 5 6 6 6 10 ··· 10 10 ··· 10 12 12 12 12 12 12 15 15 20 ··· 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 3 3 3 3 2 2 2 6 6 10 10 10 10 30 30 30 30 2 2 2 2 2 2 ··· 2 6 ··· 6 4 4 20 20 20 20 4 4 2 ··· 2 6 ··· 6 4 ··· 4 4 ··· 4

78 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + - + + + + + + + + - - + + + - image C1 C2 C2 C2 C2 C2 C2 S3 Q8 D5 D6 D6 D6 D10 D10 D10 D10 Dic10 S3×Q8 S3×D5 C2×S3×D5 C2×S3×D5 S3×Dic10 kernel C2×S3×Dic10 S3×Dic10 C2×S3×Dic5 C2×C15⋊Q8 C6×Dic10 S3×C2×C20 C2×Dic30 C2×Dic10 S3×C10 S3×C2×C4 Dic10 C2×Dic5 C2×C20 C4×S3 C2×Dic3 C2×C12 C22×S3 D6 C10 C2×C4 C4 C22 C2 # reps 1 8 2 2 1 1 1 1 4 2 4 2 1 8 2 2 2 16 2 2 4 2 8

Matrix representation of C2×S3×Dic10 in GL6(𝔽61)

 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 60 0 0 0 0 1 60
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 50 0 0 0 0 0 11 11 0 0 0 0 0 0 17 60 0 0 0 0 45 1 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 60 59 0 0 0 0 1 1 0 0 0 0 0 0 33 38 0 0 0 0 42 28 0 0 0 0 0 0 60 0 0 0 0 0 0 60

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,60,60],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[50,11,0,0,0,0,0,11,0,0,0,0,0,0,17,45,0,0,0,0,60,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[60,1,0,0,0,0,59,1,0,0,0,0,0,0,33,42,0,0,0,0,38,28,0,0,0,0,0,0,60,0,0,0,0,0,0,60] >;

C2×S3×Dic10 in GAP, Magma, Sage, TeX

C_2\times S_3\times {\rm Dic}_{10}
% in TeX

G:=Group("C2xS3xDic10");
// GroupNames label

G:=SmallGroup(480,1078);
// by ID

G=gap.SmallGroup(480,1078);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,675,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^2=d^20=1,e^2=d^10,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,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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