Copied to
clipboard

G = C3×D4×Dic5order 480 = 25·3·5

Direct product of C3, D4 and Dic5

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

Aliases: C3×D4×Dic5, C56(D4×C12), C1544(C4×D4), C6028(C2×C4), C206(C2×C12), (C5×D4)⋊6C12, C41(C6×Dic5), (D4×C15)⋊12C4, C4⋊Dic513C6, (C4×Dic5)⋊4C6, (D4×C10).4C6, (C6×D4).14D5, (D4×C30).9C2, C126(C2×Dic5), C10.37(C6×D4), C6.191(D4×D5), C23.D57C6, C30.350(C2×D4), C222(C6×Dic5), C23.21(C6×D5), (C12×Dic5)⋊16C2, (C2×C12).361D10, (C22×Dic5)⋊7C6, (C22×C6).77D10, C30.237(C4○D4), (C2×C30).366C23, (C2×C60).292C22, C30.223(C22×C4), C10.38(C22×C12), C6.119(D42D5), C6.35(C22×Dic5), (C22×C30).103C22, (C6×Dic5).249C22, C2.5(C3×D4×D5), (C2×C30)⋊28(C2×C4), (C2×C10)⋊9(C2×C12), C2.6(C2×C6×Dic5), (C2×C6×Dic5)⋊15C2, (C2×D4).7(C3×D5), (C2×C6)⋊4(C2×Dic5), (C2×C4).48(C6×D5), C22.25(D5×C2×C6), (C2×C20).29(C2×C6), (C3×C4⋊Dic5)⋊31C2, C10.27(C3×C4○D4), C2.5(C3×D42D5), (C3×C23.D5)⋊23C2, (C22×C10).22(C2×C6), (C2×C10).49(C22×C6), (C2×Dic5).63(C2×C6), (C2×C6).362(C22×D5), SmallGroup(480,727)

Series: Derived Chief Lower central Upper central

C1C10 — C3×D4×Dic5
C1C5C10C2×C10C2×C30C6×Dic5C2×C6×Dic5 — C3×D4×Dic5
C5C10 — C3×D4×Dic5
C1C2×C6C6×D4

Generators and relations for C3×D4×Dic5
 G = < a,b,c,d,e | a3=b4=c2=d10=1, e2=d5, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 464 in 188 conjugacy classes, 102 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×5], C22, C22 [×4], C22 [×4], C5, C6 [×3], C6 [×4], C2×C4, C2×C4 [×8], D4 [×4], C23 [×2], C10 [×3], C10 [×4], C12 [×2], C12 [×5], C2×C6, C2×C6 [×4], C2×C6 [×4], C15, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, Dic5 [×2], Dic5 [×3], C20 [×2], C2×C10, C2×C10 [×4], C2×C10 [×4], C2×C12, C2×C12 [×8], C3×D4 [×4], C22×C6 [×2], C30 [×3], C30 [×4], C4×D4, C2×Dic5 [×2], C2×Dic5 [×2], C2×Dic5 [×4], C2×C20, C5×D4 [×4], C22×C10 [×2], C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, C22×C12 [×2], C6×D4, C3×Dic5 [×2], C3×Dic5 [×3], C60 [×2], C2×C30, C2×C30 [×4], C2×C30 [×4], C4×Dic5, C4⋊Dic5, C23.D5 [×2], C22×Dic5 [×2], D4×C10, D4×C12, C6×Dic5 [×2], C6×Dic5 [×2], C6×Dic5 [×4], C2×C60, D4×C15 [×4], C22×C30 [×2], D4×Dic5, C12×Dic5, C3×C4⋊Dic5, C3×C23.D5 [×2], C2×C6×Dic5 [×2], D4×C30, C3×D4×Dic5
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], D4 [×2], C23, D5, C12 [×4], C2×C6 [×7], C22×C4, C2×D4, C4○D4, Dic5 [×4], D10 [×3], C2×C12 [×6], C3×D4 [×2], C22×C6, C3×D5, C4×D4, C2×Dic5 [×6], C22×D5, C22×C12, C6×D4, C3×C4○D4, C3×Dic5 [×4], C6×D5 [×3], D4×D5, D42D5, C22×Dic5, D4×C12, C6×Dic5 [×6], D5×C2×C6, D4×Dic5, C3×D4×D5, C3×D42D5, C2×C6×Dic5, C3×D4×Dic5

Smallest permutation representation of C3×D4×Dic5
On 240 points
Generators in S240
(1 54 34)(2 55 35)(3 56 36)(4 57 37)(5 58 38)(6 59 39)(7 60 40)(8 51 31)(9 52 32)(10 53 33)(11 222 202)(12 223 203)(13 224 204)(14 225 205)(15 226 206)(16 227 207)(17 228 208)(18 229 209)(19 230 210)(20 221 201)(21 61 41)(22 62 42)(23 63 43)(24 64 44)(25 65 45)(26 66 46)(27 67 47)(28 68 48)(29 69 49)(30 70 50)(71 111 91)(72 112 92)(73 113 93)(74 114 94)(75 115 95)(76 116 96)(77 117 97)(78 118 98)(79 119 99)(80 120 100)(81 128 101)(82 129 102)(83 130 103)(84 121 104)(85 122 105)(86 123 106)(87 124 107)(88 125 108)(89 126 109)(90 127 110)(131 171 151)(132 172 152)(133 173 153)(134 174 154)(135 175 155)(136 176 156)(137 177 157)(138 178 158)(139 179 159)(140 180 160)(141 181 161)(142 182 162)(143 183 163)(144 184 164)(145 185 165)(146 186 166)(147 187 167)(148 188 168)(149 189 169)(150 190 170)(191 231 211)(192 232 212)(193 233 213)(194 234 214)(195 235 215)(196 236 216)(197 237 217)(198 238 218)(199 239 219)(200 240 220)
(1 79 28 83)(2 80 29 84)(3 71 30 85)(4 72 21 86)(5 73 22 87)(6 74 23 88)(7 75 24 89)(8 76 25 90)(9 77 26 81)(10 78 27 82)(11 176 231 187)(12 177 232 188)(13 178 233 189)(14 179 234 190)(15 180 235 181)(16 171 236 182)(17 172 237 183)(18 173 238 184)(19 174 239 185)(20 175 240 186)(31 96 45 110)(32 97 46 101)(33 98 47 102)(34 99 48 103)(35 100 49 104)(36 91 50 105)(37 92 41 106)(38 93 42 107)(39 94 43 108)(40 95 44 109)(51 116 65 127)(52 117 66 128)(53 118 67 129)(54 119 68 130)(55 120 69 121)(56 111 70 122)(57 112 61 123)(58 113 62 124)(59 114 63 125)(60 115 64 126)(131 196 142 207)(132 197 143 208)(133 198 144 209)(134 199 145 210)(135 200 146 201)(136 191 147 202)(137 192 148 203)(138 193 149 204)(139 194 150 205)(140 195 141 206)(151 216 162 227)(152 217 163 228)(153 218 164 229)(154 219 165 230)(155 220 166 221)(156 211 167 222)(157 212 168 223)(158 213 169 224)(159 214 170 225)(160 215 161 226)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 236)(12 237)(13 238)(14 239)(15 240)(16 231)(17 232)(18 233)(19 234)(20 235)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)(71 90)(72 81)(73 82)(74 83)(75 84)(76 85)(77 86)(78 87)(79 88)(80 89)(91 110)(92 101)(93 102)(94 103)(95 104)(96 105)(97 106)(98 107)(99 108)(100 109)(111 127)(112 128)(113 129)(114 130)(115 121)(116 122)(117 123)(118 124)(119 125)(120 126)(131 136)(132 137)(133 138)(134 139)(135 140)(141 146)(142 147)(143 148)(144 149)(145 150)(151 156)(152 157)(153 158)(154 159)(155 160)(161 166)(162 167)(163 168)(164 169)(165 170)(171 176)(172 177)(173 178)(174 179)(175 180)(181 186)(182 187)(183 188)(184 189)(185 190)(191 207)(192 208)(193 209)(194 210)(195 201)(196 202)(197 203)(198 204)(199 205)(200 206)(211 227)(212 228)(213 229)(214 230)(215 221)(216 222)(217 223)(218 224)(219 225)(220 226)
(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)
(1 131 6 136)(2 140 7 135)(3 139 8 134)(4 138 9 133)(5 137 10 132)(11 130 16 125)(12 129 17 124)(13 128 18 123)(14 127 19 122)(15 126 20 121)(21 149 26 144)(22 148 27 143)(23 147 28 142)(24 146 29 141)(25 145 30 150)(31 154 36 159)(32 153 37 158)(33 152 38 157)(34 151 39 156)(35 160 40 155)(41 169 46 164)(42 168 47 163)(43 167 48 162)(44 166 49 161)(45 165 50 170)(51 174 56 179)(52 173 57 178)(53 172 58 177)(54 171 59 176)(55 180 60 175)(61 189 66 184)(62 188 67 183)(63 187 68 182)(64 186 69 181)(65 185 70 190)(71 194 76 199)(72 193 77 198)(73 192 78 197)(74 191 79 196)(75 200 80 195)(81 209 86 204)(82 208 87 203)(83 207 88 202)(84 206 89 201)(85 205 90 210)(91 214 96 219)(92 213 97 218)(93 212 98 217)(94 211 99 216)(95 220 100 215)(101 229 106 224)(102 228 107 223)(103 227 108 222)(104 226 109 221)(105 225 110 230)(111 234 116 239)(112 233 117 238)(113 232 118 237)(114 231 119 236)(115 240 120 235)

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

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

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

120 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B4A4B4C4D4E4F4G···4L5A5B6A···6F6G···6N10A···10F10G···10N12A12B12C12D12E···12L12M···12X15A15B15C15D20A20B20C20D30A···30L30M···30AB60A···60H
order12222222334444444···4556···66···610···1010···101212121212···1212···12151515152020202030···3030···3060···60
size111122221122555510···10221···12···22···24···422225···510···10222244442···24···44···4

120 irreducible representations

dim111111111111112222222222224444
type+++++++++-++-
imageC1C2C2C2C2C2C3C4C6C6C6C6C6C12D4D5C4○D4D10Dic5D10C3×D4C3×D5C3×C4○D4C6×D5C3×Dic5C6×D5D4×D5D42D5C3×D4×D5C3×D42D5
kernelC3×D4×Dic5C12×Dic5C3×C4⋊Dic5C3×C23.D5C2×C6×Dic5D4×C30D4×Dic5D4×C15C4×Dic5C4⋊Dic5C23.D5C22×Dic5D4×C10C5×D4C3×Dic5C6×D4C30C2×C12C3×D4C22×C6Dic5C2×D4C10C2×C4D4C23C6C6C2C2
# reps11122128224421622228444441682244

Matrix representation of C3×D4×Dic5 in GL5(𝔽61)

130000
01000
00100
000470
000047
,
600000
060000
006000
00001
000600
,
10000
01000
00100
000600
00001
,
600000
00100
0604300
000600
000060
,
110000
0255700
0343600
000500
000050

G:=sub<GL(5,GF(61))| [13,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,47,0,0,0,0,0,47],[60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,1,43,0,0,0,0,0,60,0,0,0,0,0,60],[11,0,0,0,0,0,25,34,0,0,0,57,36,0,0,0,0,0,50,0,0,0,0,0,50] >;

C3×D4×Dic5 in GAP, Magma, Sage, TeX

C_3\times D_4\times {\rm Dic}_5
% in TeX

G:=Group("C3xD4xDic5");
// GroupNames label

G:=SmallGroup(480,727);
// by ID

G=gap.SmallGroup(480,727);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,168,555,18822]);
// Polycyclic

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

׿
×
𝔽