(C4xDic5):S3 - GroupNames

G = (C₄×Dic₅)⋊S₃ order 480 = 2⁵·3·5

12^nd semidirect product of C₄×Dic₅ and S₃ acting via S₃/C₃=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic₃⋊C₄⋊₃D₅, C₅⋊₁(C₄²⋊₃S₃), (C₄×Dic₅)⋊₁₂S₃, (C₂×C₂₀).₁₈₈D₆, C₆.Dic₁₀⋊₉C₂, (C₁₂×Dic₅)⋊₂₂C₂, D₃₀⋊₄C₄.₇C₂, C₁₀.₈(C₄○D₁₂), C₆.₃₃(C₄○D₂₀), C₃₀.₄₉(C₄○D₄), (C₂×C₁₂).₂₆₁D₁₀, (C₂×C₃₀).₇₇C₂³, Dic₁₅⋊₅C₄⋊₁₂C₂, C₁₅⋊₁₂(C₄²⋊₂C₂), D₃₀⋊₃C₄.₁₂C₂, C₆.₄₁(D₄⋊₂D₅), (C₂×C₆₀).₃₈₄C₂², C₆.₁₀(Q₈⋊₂D₅), (C₂×Dic₅).₁₆₉D₆, (C₂×Dic₃).₂₆D₁₀, C₂.₁₃(C₁₂.₂₈D₁₀), C₂.₁₅(Dic₃.D₁₀), C₂.₂₅(D₆.D₁₀), (C₆×Dic₅).₁₉₁C₂², (C₂×Dic₁₅).₆₆C₂², (C₁₀×Dic₃).₄₅C₂², (C₂²×D₁₅).₂₆C₂², C₃⋊₁(C₄⋊C₄⋊D₅), (C₂×C₄).₁₂₄(S₃×D₅), C₂².₁₆₂(C₂×S₃×D₅), (C₅×Dic₃⋊C₄)⋊₂₅C₂, (C₂×C₆).₈₉(C₂²×D₅), (C₂×C₁₀).₈₉(C₂²×S₃), SmallGroup(480,463)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₃₀ — (C₄×Dic₅)⋊S₃

Chief series

C₁ — C₅ — C₁₅ — C₃₀ — C₂×C₃₀ — C₆×Dic₅ — Dic₁₅⋊₅C₄ — (C₄×Dic₅)⋊S₃

Lower central

C₁₅ — C₂×C₃₀ — (C₄×Dic₅)⋊S₃

Upper central

C₁ — C₂² — C₂×C₄

Generators and relations for (C₄×Dic₅)⋊S₃
G = < a,b,c,d,e | a⁴=b¹⁰=d³=e²=1, c²=b⁵, ab=ba, ac=ca, ad=da, eae=ab⁵, cbc^-1=ebe=b^-1, bd=db, cd=dc, ece=a²b⁵c, ede=d^-1 >

Subgroups: 652 in 120 conjugacy classes, 44 normal (all characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², C₅, S₃, C₆, C₂×C₄, C₂×C₄, C₂³, D₅, C₁₀, Dic₃, C₁₂, D₆, C₂×C₆, C₁₅, C₄², C₂²⋊C₄, C₄⋊C₄, Dic₅, C₂₀, D₁₀, C₂×C₁₀, C₂×Dic₃, C₂×Dic₃, C₂×C₁₂, C₂×C₁₂, C₂²×S₃, D₁₅, C₃₀, C₄²⋊₂C₂, C₂×Dic₅, C₂×Dic₅, C₂×C₂₀, C₂×C₂₀, C₂²×D₅, Dic₃⋊C₄, Dic₃⋊C₄, D₆⋊C₄, C₄×C₁₂, C₅×Dic₃, C₃×Dic₅, Dic₁₅, C₆₀, D₃₀, C₂×C₃₀, C₄×Dic₅, C₁₀.D₄, C₄⋊Dic₅, D₁₀⋊C₄, C₅×C₄⋊C₄, C₄²⋊₃S₃, C₆×Dic₅, C₁₀×Dic₃, C₂×Dic₁₅, C₂×C₆₀, C₂²×D₁₅, C₄⋊C₄⋊D₅, D₃₀⋊₄C₄, Dic₁₅⋊₅C₄, C₆.Dic₁₀, C₁₂×Dic₅, C₅×Dic₃⋊C₄, D₃₀⋊₃C₄, (C₄×Dic₅)⋊S₃
Quotients: C₁, C₂, C₂², S₃, C₂³, D₅, D₆, C₄○D₄, D₁₀, C₂²×S₃, C₄²⋊₂C₂, C₂²×D₅, C₄○D₁₂, S₃×D₅, C₄○D₂₀, D₄⋊₂D₅, Q₈⋊₂D₅, C₄²⋊₃S₃, C₂×S₃×D₅, C₄⋊C₄⋊D₅, D₆.D₁₀, C₁₂.₂₈D₁₀, Dic₃.D₁₀, (C₄×Dic₅)⋊S₃

Smallest permutation representation of (C₄×Dic₅)⋊S₃
►On 240 points

Generators in S₂₄₀

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

(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)

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

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

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

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

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

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

66 conjugacy classes

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

66 irreducible representations

dim	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+		+	+			+	-	+	+		+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	S₃	D₅	D₆	D₆	C₄○D₄	D₁₀	D₁₀	C₄○D₁₂	C₄○D₂₀	S₃×D₅	D₄⋊₂D₅	Q₈⋊₂D₅	C₂×S₃×D₅	D₆.D₁₀	C₁₂.₂₈D₁₀	Dic₃.D₁₀
kernel	(C₄×Dic₅)⋊S₃	D₃₀⋊₄C₄	Dic₁₅⋊₅C₄	C₆.Dic₁₀	C₁₂×Dic₅	C₅×Dic₃⋊C₄	D₃₀⋊₃C₄	C₄×Dic₅	Dic₃⋊C₄	C₂×Dic₅	C₂×C₂₀	C₃₀	C₂×Dic₃	C₂×C₁₂	C₁₀	C₆	C₂×C₄	C₆	C₆	C₂²	C₂	C₂	C₂
# reps	1	2	1	1	1	1	1	1	2	2	1	6	4	2	12	8	2	2	2	2	4	4	4

Matrix representation of (C₄×Dic₅)⋊S₃ ►in GL₆(𝔽₆₁)

50	0	0	0	0	0
0	50	0	0	0	0
0	0	11	0	0	0
0	0	0	11	0	0
0	0	0	0	1	6
0	0	0	0	0	60

1	0	0	0	0	0
0	1	0	0	0	0
0	0	43	60	0	0
0	0	1	0	0	0
0	0	0	0	60	0
0	0	0	0	0	60

9	18	0	0	0	0
43	52	0	0	0	0
0	0	36	4	0	0
0	0	27	25	0	0
0	0	0	0	50	56
0	0	0	0	0	11

60	60	0	0	0	0
1	0	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	1

1	0	0	0	0	0
60	60	0	0	0	0
0	0	1	0	0	0
0	0	43	60	0	0
0	0	0	0	1	0
0	0	0	0	20	60

G:=sub<GL(6,GF(61))| [50,0,0,0,0,0,0,50,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,6,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,43,1,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[9,43,0,0,0,0,18,52,0,0,0,0,0,0,36,27,0,0,0,0,4,25,0,0,0,0,0,0,50,0,0,0,0,0,56,11],[60,1,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,60,0,0,0,0,0,60,0,0,0,0,0,0,1,43,0,0,0,0,0,60,0,0,0,0,0,0,1,20,0,0,0,0,0,60] >;

(C₄×Dic₅)⋊S₃ in GAP, Magma, Sage, TeX

(C_4\times {\rm Dic}_5)\rtimes S_3

% in TeX

G:=Group("(C4xDic5):S3");

// GroupNames label

G:=SmallGroup(480,463);

// by ID

G=gap.SmallGroup(480,463);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,120,219,142,1356,18822]);

// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^10=d^3=e^2=1,c^2=b^5,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a*b^5,c*b*c^-1=e*b*e=b^-1,b*d=d*b,c*d=d*c,e*c*e=a^2*b^5*c,e*d*e=d^-1>;

// generators/relations

G = (C4×Dic5)⋊S3 order 480 = 25·3·5

12nd semidirect product of C4×Dic5 and S3 acting via S3/C3=C2

G = (C₄×Dic₅)⋊S₃ order 480 = 2⁵·3·5

12^nd semidirect product of C₄×Dic₅ and S₃ acting via S₃/C₃=C₂