C60.19C2^3 - GroupNames

G = C₆₀.₁₉C₂³ order 480 = 2⁵·3·5

19^th non-split extension by C₆₀ of C₂³ acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₆₀.₁₉C₂³, Dic₆.₉D₁₀, D₆₀.₅C₂², Dic₁₀.₂₄D₆, D₄⋊D₁₅⋊₇C₂, D₄.D₅⋊₇S₃, D₄.₅(S₃×D₅), D₄⋊₂S₃⋊₃D₅, C₁₅⋊₁₆(C₄○D₈), (C₅×D₄).₂₂D₆, C₁₅⋊Q₁₆⋊₄C₂, (C₃×D₄).₇D₁₀, C₅⋊₂C₈.₁₄D₆, D₆₀⋊C₂⋊₁C₂, (S₃×C₁₀).₁₁D₄, (C₄×S₃).₂₂D₁₀, C₁₀.₁₄₅(S₃×D₄), C₃₀.₁₈₁(C₂×D₄), Dic₆⋊D₅⋊₃C₂, D₆.₂(C₅⋊D₄), C₅⋊₆(Q₈.₇D₆), C₃⋊₃(D₄.₈D₁₀), (S₃×C₂₀).₇C₂², C₂₀.₁₉(C₂²×S₃), C₁₅⋊₃C₈.₆C₂², (C₅×Dic₃).₃₇D₄, C₁₂.₁₉(C₂²×D₅), (D₄×C₁₅).₁₃C₂², (C₅×Dic₆).₆C₂², Dic₃.₂₁(C₅⋊D₄), (C₃×Dic₁₀).₆C₂², C₄.₁₉(C₂×S₃×D₅), (S₃×C₅⋊₂C₈)⋊₄C₂, (C₃×D₄.D₅)⋊₅C₂, C₂.₂₆(S₃×C₅⋊D₄), C₆.₄₈(C₂×C₅⋊D₄), (C₅×D₄⋊₂S₃)⋊₃C₂, (C₃×C₅⋊₂C₈).₄C₂², SmallGroup(480,571)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆₀ — C₆₀.₁₉C₂³

Chief series

C₁ — C₅ — C₁₅ — C₃₀ — C₆₀ — C₃×Dic₁₀ — D₆₀⋊C₂ — C₆₀.₁₉C₂³

Lower central

C₁₅ — C₃₀ — C₆₀ — C₆₀.₁₉C₂³

Upper central

C₁ — C₂ — C₄ — D₄

Generators and relations for C₆₀.₁₉C₂³
G = < a,b,c,d | a⁶⁰=d²=1, b²=c²=a³⁰, bab^-1=a¹⁹, cac^-1=a⁴¹, dad=a³¹, bc=cb, dbd=a⁴⁵b, cd=dc >

Subgroups: 636 in 124 conjugacy classes, 40 normal (all characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₅, S₃, C₆, C₆, C₈, C₂×C₄, D₄, D₄, Q₈, D₅, C₁₀, C₁₀, Dic₃, Dic₃, C₁₂, C₁₂, D₆, D₆, C₂×C₆, C₁₅, C₂×C₈, D₈, SD₁₆, Q₁₆, C₄○D₄, Dic₅, C₂₀, C₂₀, D₁₀, C₂×C₁₀, C₃⋊C₈, C₂₄, Dic₆, C₄×S₃, C₄×S₃, D₁₂, C₂×Dic₃, C₃⋊D₄, C₃×D₄, C₃×Q₈, C₅×S₃, D₁₅, C₃₀, C₃₀, C₄○D₈, C₅⋊₂C₈, C₅⋊₂C₈, Dic₁₀, C₄×D₅, D₂₀, C₅⋊D₄, C₂×C₂₀, C₅×D₄, C₅×D₄, C₅×Q₈, S₃×C₈, C₂₄⋊C₂, D₄⋊S₃, C₃⋊Q₁₆, C₃×SD₁₆, D₄⋊₂S₃, Q₈⋊₃S₃, C₅×Dic₃, C₅×Dic₃, C₃×Dic₅, C₆₀, S₃×C₁₀, D₃₀, C₂×C₃₀, C₂×C₅⋊₂C₈, D₄⋊D₅, D₄.D₅, Q₈⋊D₅, C₅⋊Q₁₆, C₄○D₂₀, C₅×C₄○D₄, Q₈.₇D₆, C₃×C₅⋊₂C₈, C₁₅⋊₃C₈, D₃₀.C₂, C₅⋊D₁₂, C₃×Dic₁₀, C₅×Dic₆, S₃×C₂₀, C₁₀×Dic₃, C₅×C₃⋊D₄, D₆₀, D₄×C₁₅, D₄.₈D₁₀, S₃×C₅⋊₂C₈, Dic₆⋊D₅, C₁₅⋊Q₁₆, C₃×D₄.D₅, D₄⋊D₁₅, D₆₀⋊C₂, C₅×D₄⋊₂S₃, C₆₀.₁₉C₂³
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₅, D₆, C₂×D₄, D₁₀, C₂²×S₃, C₄○D₈, C₅⋊D₄, C₂²×D₅, S₃×D₄, S₃×D₅, C₂×C₅⋊D₄, Q₈.₇D₆, C₂×S₃×D₅, D₄.₈D₁₀, S₃×C₅⋊D₄, C₆₀.₁₉C₂³

Smallest permutation representation of C₆₀.₁₉C₂³
►On 240 points

Generators in S₂₄₀

(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)

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

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

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

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

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

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

51 conjugacy classes

class	1	2A	2B	2C	2D	3	4A	4B	4C	4D	4E	5A	5B	6A	6B	8A	8B	8C	8D	10A	10B	10C	10D	10E	10F	10G	10H	12A	12B	15A	15B	20A	20B	20C	20D	20E	20F	20G	20H	20I	20J	24A	24B	30A	30B	30C	30D	30E	30F	60A	60B
order	1	2	2	2	2	3	4	4	4	4	4	5	5	6	6	8	8	8	8	10	10	10	10	10	10	10	10	12	12	15	15	20	20	20	20	20	20	20	20	20	20	24	24	30	30	30	30	30	30	60	60
size	1	1	4	6	60	2	2	3	3	12	20	2	2	2	8	10	10	30	30	2	2	4	4	4	4	12	12	4	40	4	4	4	4	6	6	6	6	12	12	12	12	20	20	4	4	8	8	8	8	8	8

51 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4	8
type	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+				+	+		+			+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	S₃	D₄	D₄	D₅	D₆	D₆	D₆	D₁₀	D₁₀	D₁₀	C₄○D₈	C₅⋊D₄	C₅⋊D₄	S₃×D₄	S₃×D₅	Q₈.₇D₆	C₂×S₃×D₅	D₄.₈D₁₀	S₃×C₅⋊D₄	C₆₀.₁₉C₂³
kernel	C₆₀.₁₉C₂³	S₃×C₅⋊₂C₈	Dic₆⋊D₅	C₁₅⋊Q₁₆	C₃×D₄.D₅	D₄⋊D₁₅	D₆₀⋊C₂	C₅×D₄⋊₂S₃	D₄.D₅	C₅×Dic₃	S₃×C₁₀	D₄⋊₂S₃	C₅⋊₂C₈	Dic₁₀	C₅×D₄	Dic₆	C₄×S₃	C₃×D₄	C₁₅	Dic₃	D₆	C₁₀	D₄	C₅	C₄	C₃	C₂	C₁
# reps	1	1	1	1	1	1	1	1	1	1	1	2	1	1	1	2	2	2	4	4	4	1	2	2	2	4	4	2

Matrix representation of C₆₀.₁₉C₂³ ►in GL₆(𝔽₂₄₁)

52	52	0	0	0	0
189	240	0	0	0	0
0	0	1	54	0	0
0	0	174	239	0	0
0	0	0	0	177	0
0	0	0	0	18	64

40	146	0	0	0	0
235	201	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	154	24
0	0	0	0	26	87

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	174	240	0	0
0	0	0	0	64	0
0	0	0	0	0	64

1	0	0	0	0	0
0	1	0	0	0	0
0	0	240	0	0	0
0	0	0	240	0	0
0	0	0	0	200	3
0	0	0	0	163	41

G:=sub<GL(6,GF(241))| [52,189,0,0,0,0,52,240,0,0,0,0,0,0,1,174,0,0,0,0,54,239,0,0,0,0,0,0,177,18,0,0,0,0,0,64],[40,235,0,0,0,0,146,201,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,154,26,0,0,0,0,24,87],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,174,0,0,0,0,0,240,0,0,0,0,0,0,64,0,0,0,0,0,0,64],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,200,163,0,0,0,0,3,41] >;

C₆₀.₁₉C₂³ in GAP, Magma, Sage, TeX

C_{60}._{19}C_2^3

% in TeX

G:=Group("C60.19C2^3");

// GroupNames label

G:=SmallGroup(480,571);

// by ID

G=gap.SmallGroup(480,571);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,120,254,675,185,80,1356,18822]);

// Polycyclic

G:=Group<a,b,c,d|a^60=d^2=1,b^2=c^2=a^30,b*a*b^-1=a^19,c*a*c^-1=a^41,d*a*d=a^31,b*c=c*b,d*b*d=a^45*b,c*d=d*c>;

// generators/relations

G = C60.19C23 order 480 = 25·3·5

19th non-split extension by C60 of C23 acting faithfully

G = C₆₀.₁₉C₂³ order 480 = 2⁵·3·5

19^th non-split extension by C₆₀ of C₂³ acting faithfully