direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×D4×Dic5, C5⋊6(D4×C12), C15⋊44(C4×D4), C60⋊28(C2×C4), C20⋊6(C2×C12), (C5×D4)⋊6C12, C4⋊1(C6×Dic5), (D4×C15)⋊12C4, C4⋊Dic5⋊13C6, (C4×Dic5)⋊4C6, (D4×C10).4C6, (C6×D4).14D5, (D4×C30).9C2, C12⋊6(C2×Dic5), C10.37(C6×D4), C6.191(D4×D5), C23.D5⋊7C6, C30.350(C2×D4), C22⋊2(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(D4⋊2D5), 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×D4⋊2D5), (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
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, C2, C3, C4, C4, C22, C22, C22, C5, C6, C6, C2×C4, C2×C4, D4, C23, C10, C10, C12, C12, C2×C6, C2×C6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, Dic5, C20, C2×C10, C2×C10, C2×C10, C2×C12, C2×C12, C3×D4, C22×C6, C30, C30, C4×D4, C2×Dic5, C2×Dic5, C2×Dic5, C2×C20, C5×D4, C22×C10, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C3×Dic5, C3×Dic5, C60, C2×C30, C2×C30, C2×C30, C4×Dic5, C4⋊Dic5, C23.D5, C22×Dic5, D4×C10, D4×C12, C6×Dic5, C6×Dic5, C6×Dic5, C2×C60, D4×C15, C22×C30, D4×Dic5, C12×Dic5, C3×C4⋊Dic5, C3×C23.D5, C2×C6×Dic5, D4×C30, C3×D4×Dic5
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C23, D5, C12, C2×C6, C22×C4, C2×D4, C4○D4, Dic5, D10, C2×C12, C3×D4, C22×C6, C3×D5, C4×D4, C2×Dic5, C22×D5, C22×C12, C6×D4, C3×C4○D4, C3×Dic5, C6×D5, D4×D5, D4⋊2D5, C22×Dic5, D4×C12, C6×Dic5, D5×C2×C6, D4×Dic5, C3×D4×D5, C3×D4⋊2D5, C2×C6×Dic5, C3×D4×Dic5
►On 240 points
Generators in S240
(1 58 38)(2 59 39)(3 60 40)(4 51 31)(5 52 32)(6 53 33)(7 54 34)(8 55 35)(9 56 36)(10 57 37)(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 73 28 83)(2 74 29 84)(3 75 30 85)(4 76 21 86)(5 77 22 87)(6 78 23 88)(7 79 24 89)(8 80 25 90)(9 71 26 81)(10 72 27 82)(11 172 237 187)(12 173 238 188)(13 174 239 189)(14 175 240 190)(15 176 231 181)(16 177 232 182)(17 178 233 183)(18 179 234 184)(19 180 235 185)(20 171 236 186)(31 96 41 106)(32 97 42 107)(33 98 43 108)(34 99 44 109)(35 100 45 110)(36 91 46 101)(37 92 47 102)(38 93 48 103)(39 94 49 104)(40 95 50 105)(51 116 61 123)(52 117 62 124)(53 118 63 125)(54 119 64 126)(55 120 65 127)(56 111 66 128)(57 112 67 129)(58 113 68 130)(59 114 69 121)(60 115 70 122)(131 196 146 201)(132 197 147 202)(133 198 148 203)(134 199 149 204)(135 200 150 205)(136 191 141 206)(137 192 142 207)(138 193 143 208)(139 194 144 209)(140 195 145 210)(151 216 166 221)(152 217 167 222)(153 218 168 223)(154 219 169 224)(155 220 170 225)(156 211 161 226)(157 212 162 227)(158 213 163 228)(159 214 164 229)(160 215 165 230)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 232)(12 233)(13 234)(14 235)(15 236)(16 237)(17 238)(18 239)(19 240)(20 231)(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 86)(72 87)(73 88)(74 89)(75 90)(76 81)(77 82)(78 83)(79 84)(80 85)(91 106)(92 107)(93 108)(94 109)(95 110)(96 101)(97 102)(98 103)(99 104)(100 105)(111 123)(112 124)(113 125)(114 126)(115 127)(116 128)(117 129)(118 130)(119 121)(120 122)(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)(191 201)(192 202)(193 203)(194 204)(195 205)(196 206)(197 207)(198 208)(199 209)(200 210)(211 221)(212 222)(213 223)(214 224)(215 225)(216 226)(217 227)(218 228)(219 229)(220 230)
(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)
(1 137 6 132)(2 136 7 131)(3 135 8 140)(4 134 9 139)(5 133 10 138)(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)| (1,58,38)(2,59,39)(3,60,40)(4,51,31)(5,52,32)(6,53,33)(7,54,34)(8,55,35)(9,56,36)(10,57,37)(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), (1,73,28,83)(2,74,29,84)(3,75,30,85)(4,76,21,86)(5,77,22,87)(6,78,23,88)(7,79,24,89)(8,80,25,90)(9,71,26,81)(10,72,27,82)(11,172,237,187)(12,173,238,188)(13,174,239,189)(14,175,240,190)(15,176,231,181)(16,177,232,182)(17,178,233,183)(18,179,234,184)(19,180,235,185)(20,171,236,186)(31,96,41,106)(32,97,42,107)(33,98,43,108)(34,99,44,109)(35,100,45,110)(36,91,46,101)(37,92,47,102)(38,93,48,103)(39,94,49,104)(40,95,50,105)(51,116,61,123)(52,117,62,124)(53,118,63,125)(54,119,64,126)(55,120,65,127)(56,111,66,128)(57,112,67,129)(58,113,68,130)(59,114,69,121)(60,115,70,122)(131,196,146,201)(132,197,147,202)(133,198,148,203)(134,199,149,204)(135,200,150,205)(136,191,141,206)(137,192,142,207)(138,193,143,208)(139,194,144,209)(140,195,145,210)(151,216,166,221)(152,217,167,222)(153,218,168,223)(154,219,169,224)(155,220,170,225)(156,211,161,226)(157,212,162,227)(158,213,163,228)(159,214,164,229)(160,215,165,230), (1,6)(2,7)(3,8)(4,9)(5,10)(11,232)(12,233)(13,234)(14,235)(15,236)(16,237)(17,238)(18,239)(19,240)(20,231)(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)(71,86)(72,87)(73,88)(74,89)(75,90)(76,81)(77,82)(78,83)(79,84)(80,85)(91,106)(92,107)(93,108)(94,109)(95,110)(96,101)(97,102)(98,103)(99,104)(100,105)(111,123)(112,124)(113,125)(114,126)(115,127)(116,128)(117,129)(118,130)(119,121)(120,122)(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)(191,201)(192,202)(193,203)(194,204)(195,205)(196,206)(197,207)(198,208)(199,209)(200,210)(211,221)(212,222)(213,223)(214,224)(215,225)(216,226)(217,227)(218,228)(219,229)(220,230), (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), (1,137,6,132)(2,136,7,131)(3,135,8,140)(4,134,9,139)(5,133,10,138)(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( (1,58,38)(2,59,39)(3,60,40)(4,51,31)(5,52,32)(6,53,33)(7,54,34)(8,55,35)(9,56,36)(10,57,37)(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), (1,73,28,83)(2,74,29,84)(3,75,30,85)(4,76,21,86)(5,77,22,87)(6,78,23,88)(7,79,24,89)(8,80,25,90)(9,71,26,81)(10,72,27,82)(11,172,237,187)(12,173,238,188)(13,174,239,189)(14,175,240,190)(15,176,231,181)(16,177,232,182)(17,178,233,183)(18,179,234,184)(19,180,235,185)(20,171,236,186)(31,96,41,106)(32,97,42,107)(33,98,43,108)(34,99,44,109)(35,100,45,110)(36,91,46,101)(37,92,47,102)(38,93,48,103)(39,94,49,104)(40,95,50,105)(51,116,61,123)(52,117,62,124)(53,118,63,125)(54,119,64,126)(55,120,65,127)(56,111,66,128)(57,112,67,129)(58,113,68,130)(59,114,69,121)(60,115,70,122)(131,196,146,201)(132,197,147,202)(133,198,148,203)(134,199,149,204)(135,200,150,205)(136,191,141,206)(137,192,142,207)(138,193,143,208)(139,194,144,209)(140,195,145,210)(151,216,166,221)(152,217,167,222)(153,218,168,223)(154,219,169,224)(155,220,170,225)(156,211,161,226)(157,212,162,227)(158,213,163,228)(159,214,164,229)(160,215,165,230), (1,6)(2,7)(3,8)(4,9)(5,10)(11,232)(12,233)(13,234)(14,235)(15,236)(16,237)(17,238)(18,239)(19,240)(20,231)(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)(71,86)(72,87)(73,88)(74,89)(75,90)(76,81)(77,82)(78,83)(79,84)(80,85)(91,106)(92,107)(93,108)(94,109)(95,110)(96,101)(97,102)(98,103)(99,104)(100,105)(111,123)(112,124)(113,125)(114,126)(115,127)(116,128)(117,129)(118,130)(119,121)(120,122)(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)(191,201)(192,202)(193,203)(194,204)(195,205)(196,206)(197,207)(198,208)(199,209)(200,210)(211,221)(212,222)(213,223)(214,224)(215,225)(216,226)(217,227)(218,228)(219,229)(220,230), (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), (1,137,6,132)(2,136,7,131)(3,135,8,140)(4,134,9,139)(5,133,10,138)(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([[(1,58,38),(2,59,39),(3,60,40),(4,51,31),(5,52,32),(6,53,33),(7,54,34),(8,55,35),(9,56,36),(10,57,37),(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)], [(1,73,28,83),(2,74,29,84),(3,75,30,85),(4,76,21,86),(5,77,22,87),(6,78,23,88),(7,79,24,89),(8,80,25,90),(9,71,26,81),(10,72,27,82),(11,172,237,187),(12,173,238,188),(13,174,239,189),(14,175,240,190),(15,176,231,181),(16,177,232,182),(17,178,233,183),(18,179,234,184),(19,180,235,185),(20,171,236,186),(31,96,41,106),(32,97,42,107),(33,98,43,108),(34,99,44,109),(35,100,45,110),(36,91,46,101),(37,92,47,102),(38,93,48,103),(39,94,49,104),(40,95,50,105),(51,116,61,123),(52,117,62,124),(53,118,63,125),(54,119,64,126),(55,120,65,127),(56,111,66,128),(57,112,67,129),(58,113,68,130),(59,114,69,121),(60,115,70,122),(131,196,146,201),(132,197,147,202),(133,198,148,203),(134,199,149,204),(135,200,150,205),(136,191,141,206),(137,192,142,207),(138,193,143,208),(139,194,144,209),(140,195,145,210),(151,216,166,221),(152,217,167,222),(153,218,168,223),(154,219,169,224),(155,220,170,225),(156,211,161,226),(157,212,162,227),(158,213,163,228),(159,214,164,229),(160,215,165,230)], [(1,6),(2,7),(3,8),(4,9),(5,10),(11,232),(12,233),(13,234),(14,235),(15,236),(16,237),(17,238),(18,239),(19,240),(20,231),(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),(71,86),(72,87),(73,88),(74,89),(75,90),(76,81),(77,82),(78,83),(79,84),(80,85),(91,106),(92,107),(93,108),(94,109),(95,110),(96,101),(97,102),(98,103),(99,104),(100,105),(111,123),(112,124),(113,125),(114,126),(115,127),(116,128),(117,129),(118,130),(119,121),(120,122),(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),(191,201),(192,202),(193,203),(194,204),(195,205),(196,206),(197,207),(198,208),(199,209),(200,210),(211,221),(212,222),(213,223),(214,224),(215,225),(216,226),(217,227),(218,228),(219,229),(220,230)], [(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)], [(1,137,6,132),(2,136,7,131),(3,135,8,140),(4,134,9,139),(5,133,10,138),(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 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 5A | 5B | 6A | ··· | 6F | 6G | ··· | 6N | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 12M | ··· | 12X | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 30A | ··· | 30L | 30M | ··· | 30AB | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 6 | ··· | 6 | 6 | ··· | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 5 | ··· | 5 | 10 | ··· | 10 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | - | |||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C6 | C12 | D4 | D5 | C4○D4 | D10 | Dic5 | D10 | C3×D4 | C3×D5 | C3×C4○D4 | C6×D5 | C3×Dic5 | C6×D5 | D4×D5 | D4⋊2D5 | C3×D4×D5 | C3×D4⋊2D5 |
| kernel | C3×D4×Dic5 | C12×Dic5 | C3×C4⋊Dic5 | C3×C23.D5 | C2×C6×Dic5 | D4×C30 | D4×Dic5 | D4×C15 | C4×Dic5 | C4⋊Dic5 | C23.D5 | C22×Dic5 | D4×C10 | C5×D4 | C3×Dic5 | C6×D4 | C30 | C2×C12 | C3×D4 | C22×C6 | Dic5 | C2×D4 | C10 | C2×C4 | D4 | C23 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 8 | 2 | 2 | 4 | 4 | 2 | 16 | 2 | 2 | 2 | 2 | 8 | 4 | 4 | 4 | 4 | 4 | 16 | 8 | 2 | 2 | 4 | 4 |
Matrix representation of C3×D4×Dic5 ►in GL5(𝔽61)
| 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 | 1 |
| 0 | 0 | 0 | 60 | 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 | 1 | 0 | 0 |
| 0 | 60 | 43 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 60 |
| 11 | 0 | 0 | 0 | 0 |
| 0 | 25 | 57 | 0 | 0 |
| 0 | 34 | 36 | 0 | 0 |
| 0 | 0 | 0 | 50 | 0 |
| 0 | 0 | 0 | 0 | 50 |
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