metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C4×Dic3)⋊10D5, (C2×C20).259D6, C3⋊2(C42⋊2D5), Dic15⋊5C4⋊7C2, C30.Q8⋊5C2, (C22×D5).8D6, C15⋊7(C42⋊2C2), (Dic3×C20)⋊21C2, D10⋊C4.1S3, C30.29(C4○D4), C6.66(C4○D20), (C2×C12).180D10, (C2×C30).53C23, C30.4Q8⋊29C2, (C2×Dic5).12D6, C5⋊1(C23.8D6), C10.27(C4○D12), C2.9(D20⋊5S3), (C2×C60).403C22, D10⋊Dic3.7C2, C10.21(D4⋊2S3), (C2×Dic3).138D10, (C6×Dic5).31C22, C2.16(D6.D10), C2.14(Dic5.D6), (C2×Dic15).55C22, (C10×Dic3).162C22, (C2×C4).71(S3×D5), (D5×C2×C6).6C22, C22.140(C2×S3×D5), (C2×C6).65(C22×D5), (C2×C10).65(C22×S3), (C3×D10⋊C4).14C2, SmallGroup(480,439)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C4×Dic3)⋊D5
G = < a,b,c,d,e | a4=b6=d5=e2=1, c2=b3, ab=ba, ac=ca, ad=da, eae=ab3, cbc-1=b-1, bd=db, be=eb, cd=dc, ece=a2b3c, ede=d-1 >
Subgroups: 556 in 120 conjugacy classes, 44 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, C23, D5, C10, Dic3, C12, C2×C6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, Dic5, C20, D10, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C3×D5, C30, C42⋊2C2, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C6.D4, C3×C22⋊C4, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, C2×C30, C10.D4, D10⋊C4, D10⋊C4, C4×C20, C23.8D6, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, D5×C2×C6, C42⋊2D5, D10⋊Dic3, C30.Q8, Dic15⋊5C4, C3×D10⋊C4, Dic3×C20, C30.4Q8, (C4×Dic3)⋊D5
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, D10, C22×S3, C42⋊2C2, C22×D5, C4○D12, D4⋊2S3, S3×D5, C4○D20, C23.8D6, C2×S3×D5, C42⋊2D5, D20⋊5S3, D6.D10, Dic5.D6, (C4×Dic3)⋊D5
►On 240 points
Generators in S240
(1 119 59 86)(2 120 60 87)(3 115 55 88)(4 116 56 89)(5 117 57 90)(6 118 58 85)(7 202 229 169)(8 203 230 170)(9 204 231 171)(10 199 232 172)(11 200 233 173)(12 201 234 174)(13 198 225 165)(14 193 226 166)(15 194 227 167)(16 195 228 168)(17 196 223 163)(18 197 224 164)(19 188 221 161)(20 189 222 162)(21 190 217 157)(22 191 218 158)(23 192 219 159)(24 187 220 160)(25 184 211 151)(26 185 212 152)(27 186 213 153)(28 181 214 154)(29 182 215 155)(30 183 216 156)(31 129 64 91)(32 130 65 92)(33 131 66 93)(34 132 61 94)(35 127 62 95)(36 128 63 96)(37 125 70 97)(38 126 71 98)(39 121 72 99)(40 122 67 100)(41 123 68 101)(42 124 69 102)(43 136 76 103)(44 137 77 104)(45 138 78 105)(46 133 73 106)(47 134 74 107)(48 135 75 108)(49 142 82 109)(50 143 83 110)(51 144 84 111)(52 139 79 112)(53 140 80 113)(54 141 81 114)(145 238 178 205)(146 239 179 206)(147 240 180 207)(148 235 175 208)(149 236 176 209)(150 237 177 210)
(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)
(1 178 4 175)(2 177 5 180)(3 176 6 179)(7 111 10 114)(8 110 11 113)(9 109 12 112)(13 105 16 108)(14 104 17 107)(15 103 18 106)(19 94 22 91)(20 93 23 96)(21 92 24 95)(25 98 28 101)(26 97 29 100)(27 102 30 99)(31 188 34 191)(32 187 35 190)(33 192 36 189)(37 182 40 185)(38 181 41 184)(39 186 42 183)(43 197 46 194)(44 196 47 193)(45 195 48 198)(49 201 52 204)(50 200 53 203)(51 199 54 202)(55 149 58 146)(56 148 59 145)(57 147 60 150)(61 158 64 161)(62 157 65 160)(63 162 66 159)(67 152 70 155)(68 151 71 154)(69 156 72 153)(73 167 76 164)(74 166 77 163)(75 165 78 168)(79 171 82 174)(80 170 83 173)(81 169 84 172)(85 239 88 236)(86 238 89 235)(87 237 90 240)(115 209 118 206)(116 208 119 205)(117 207 120 210)(121 213 124 216)(122 212 125 215)(123 211 126 214)(127 217 130 220)(128 222 131 219)(129 221 132 218)(133 227 136 224)(134 226 137 223)(135 225 138 228)(139 231 142 234)(140 230 143 233)(141 229 144 232)
(1 36 39 53 43)(2 31 40 54 44)(3 32 41 49 45)(4 33 42 50 46)(5 34 37 51 47)(6 35 38 52 48)(7 17 237 19 26)(8 18 238 20 27)(9 13 239 21 28)(10 14 240 22 29)(11 15 235 23 30)(12 16 236 24 25)(55 65 68 82 78)(56 66 69 83 73)(57 61 70 84 74)(58 62 71 79 75)(59 63 72 80 76)(60 64 67 81 77)(85 95 98 112 108)(86 96 99 113 103)(87 91 100 114 104)(88 92 101 109 105)(89 93 102 110 106)(90 94 97 111 107)(115 130 123 142 138)(116 131 124 143 133)(117 132 125 144 134)(118 127 126 139 135)(119 128 121 140 136)(120 129 122 141 137)(145 162 153 170 164)(146 157 154 171 165)(147 158 155 172 166)(148 159 156 173 167)(149 160 151 174 168)(150 161 152 169 163)(175 192 183 200 194)(176 187 184 201 195)(177 188 185 202 196)(178 189 186 203 197)(179 190 181 204 198)(180 191 182 199 193)(205 222 213 230 224)(206 217 214 231 225)(207 218 215 232 226)(208 219 216 233 227)(209 220 211 234 228)(210 221 212 229 223)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 221)(8 222)(9 217)(10 218)(11 219)(12 220)(13 206)(14 207)(15 208)(16 209)(17 210)(18 205)(19 229)(20 230)(21 231)(22 232)(23 233)(24 234)(25 211)(26 212)(27 213)(28 214)(29 215)(30 216)(31 54)(32 49)(33 50)(34 51)(35 52)(36 53)(55 78)(56 73)(57 74)(58 75)(59 76)(60 77)(61 84)(62 79)(63 80)(64 81)(65 82)(66 83)(85 105)(86 106)(87 107)(88 108)(89 103)(90 104)(91 111)(92 112)(93 113)(94 114)(95 109)(96 110)(97 100)(98 101)(99 102)(115 135)(116 136)(117 137)(118 138)(119 133)(120 134)(121 124)(122 125)(123 126)(127 142)(128 143)(129 144)(130 139)(131 140)(132 141)(145 194)(146 195)(147 196)(148 197)(149 198)(150 193)(151 181)(152 182)(153 183)(154 184)(155 185)(156 186)(157 201)(158 202)(159 203)(160 204)(161 199)(162 200)(163 180)(164 175)(165 176)(166 177)(167 178)(168 179)(169 191)(170 192)(171 187)(172 188)(173 189)(174 190)(223 237)(224 238)(225 239)(226 240)(227 235)(228 236)
G:=sub<Sym(240)| (1,119,59,86)(2,120,60,87)(3,115,55,88)(4,116,56,89)(5,117,57,90)(6,118,58,85)(7,202,229,169)(8,203,230,170)(9,204,231,171)(10,199,232,172)(11,200,233,173)(12,201,234,174)(13,198,225,165)(14,193,226,166)(15,194,227,167)(16,195,228,168)(17,196,223,163)(18,197,224,164)(19,188,221,161)(20,189,222,162)(21,190,217,157)(22,191,218,158)(23,192,219,159)(24,187,220,160)(25,184,211,151)(26,185,212,152)(27,186,213,153)(28,181,214,154)(29,182,215,155)(30,183,216,156)(31,129,64,91)(32,130,65,92)(33,131,66,93)(34,132,61,94)(35,127,62,95)(36,128,63,96)(37,125,70,97)(38,126,71,98)(39,121,72,99)(40,122,67,100)(41,123,68,101)(42,124,69,102)(43,136,76,103)(44,137,77,104)(45,138,78,105)(46,133,73,106)(47,134,74,107)(48,135,75,108)(49,142,82,109)(50,143,83,110)(51,144,84,111)(52,139,79,112)(53,140,80,113)(54,141,81,114)(145,238,178,205)(146,239,179,206)(147,240,180,207)(148,235,175,208)(149,236,176,209)(150,237,177,210), (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), (1,178,4,175)(2,177,5,180)(3,176,6,179)(7,111,10,114)(8,110,11,113)(9,109,12,112)(13,105,16,108)(14,104,17,107)(15,103,18,106)(19,94,22,91)(20,93,23,96)(21,92,24,95)(25,98,28,101)(26,97,29,100)(27,102,30,99)(31,188,34,191)(32,187,35,190)(33,192,36,189)(37,182,40,185)(38,181,41,184)(39,186,42,183)(43,197,46,194)(44,196,47,193)(45,195,48,198)(49,201,52,204)(50,200,53,203)(51,199,54,202)(55,149,58,146)(56,148,59,145)(57,147,60,150)(61,158,64,161)(62,157,65,160)(63,162,66,159)(67,152,70,155)(68,151,71,154)(69,156,72,153)(73,167,76,164)(74,166,77,163)(75,165,78,168)(79,171,82,174)(80,170,83,173)(81,169,84,172)(85,239,88,236)(86,238,89,235)(87,237,90,240)(115,209,118,206)(116,208,119,205)(117,207,120,210)(121,213,124,216)(122,212,125,215)(123,211,126,214)(127,217,130,220)(128,222,131,219)(129,221,132,218)(133,227,136,224)(134,226,137,223)(135,225,138,228)(139,231,142,234)(140,230,143,233)(141,229,144,232), (1,36,39,53,43)(2,31,40,54,44)(3,32,41,49,45)(4,33,42,50,46)(5,34,37,51,47)(6,35,38,52,48)(7,17,237,19,26)(8,18,238,20,27)(9,13,239,21,28)(10,14,240,22,29)(11,15,235,23,30)(12,16,236,24,25)(55,65,68,82,78)(56,66,69,83,73)(57,61,70,84,74)(58,62,71,79,75)(59,63,72,80,76)(60,64,67,81,77)(85,95,98,112,108)(86,96,99,113,103)(87,91,100,114,104)(88,92,101,109,105)(89,93,102,110,106)(90,94,97,111,107)(115,130,123,142,138)(116,131,124,143,133)(117,132,125,144,134)(118,127,126,139,135)(119,128,121,140,136)(120,129,122,141,137)(145,162,153,170,164)(146,157,154,171,165)(147,158,155,172,166)(148,159,156,173,167)(149,160,151,174,168)(150,161,152,169,163)(175,192,183,200,194)(176,187,184,201,195)(177,188,185,202,196)(178,189,186,203,197)(179,190,181,204,198)(180,191,182,199,193)(205,222,213,230,224)(206,217,214,231,225)(207,218,215,232,226)(208,219,216,233,227)(209,220,211,234,228)(210,221,212,229,223), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,221)(8,222)(9,217)(10,218)(11,219)(12,220)(13,206)(14,207)(15,208)(16,209)(17,210)(18,205)(19,229)(20,230)(21,231)(22,232)(23,233)(24,234)(25,211)(26,212)(27,213)(28,214)(29,215)(30,216)(31,54)(32,49)(33,50)(34,51)(35,52)(36,53)(55,78)(56,73)(57,74)(58,75)(59,76)(60,77)(61,84)(62,79)(63,80)(64,81)(65,82)(66,83)(85,105)(86,106)(87,107)(88,108)(89,103)(90,104)(91,111)(92,112)(93,113)(94,114)(95,109)(96,110)(97,100)(98,101)(99,102)(115,135)(116,136)(117,137)(118,138)(119,133)(120,134)(121,124)(122,125)(123,126)(127,142)(128,143)(129,144)(130,139)(131,140)(132,141)(145,194)(146,195)(147,196)(148,197)(149,198)(150,193)(151,181)(152,182)(153,183)(154,184)(155,185)(156,186)(157,201)(158,202)(159,203)(160,204)(161,199)(162,200)(163,180)(164,175)(165,176)(166,177)(167,178)(168,179)(169,191)(170,192)(171,187)(172,188)(173,189)(174,190)(223,237)(224,238)(225,239)(226,240)(227,235)(228,236)>;
G:=Group( (1,119,59,86)(2,120,60,87)(3,115,55,88)(4,116,56,89)(5,117,57,90)(6,118,58,85)(7,202,229,169)(8,203,230,170)(9,204,231,171)(10,199,232,172)(11,200,233,173)(12,201,234,174)(13,198,225,165)(14,193,226,166)(15,194,227,167)(16,195,228,168)(17,196,223,163)(18,197,224,164)(19,188,221,161)(20,189,222,162)(21,190,217,157)(22,191,218,158)(23,192,219,159)(24,187,220,160)(25,184,211,151)(26,185,212,152)(27,186,213,153)(28,181,214,154)(29,182,215,155)(30,183,216,156)(31,129,64,91)(32,130,65,92)(33,131,66,93)(34,132,61,94)(35,127,62,95)(36,128,63,96)(37,125,70,97)(38,126,71,98)(39,121,72,99)(40,122,67,100)(41,123,68,101)(42,124,69,102)(43,136,76,103)(44,137,77,104)(45,138,78,105)(46,133,73,106)(47,134,74,107)(48,135,75,108)(49,142,82,109)(50,143,83,110)(51,144,84,111)(52,139,79,112)(53,140,80,113)(54,141,81,114)(145,238,178,205)(146,239,179,206)(147,240,180,207)(148,235,175,208)(149,236,176,209)(150,237,177,210), (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), (1,178,4,175)(2,177,5,180)(3,176,6,179)(7,111,10,114)(8,110,11,113)(9,109,12,112)(13,105,16,108)(14,104,17,107)(15,103,18,106)(19,94,22,91)(20,93,23,96)(21,92,24,95)(25,98,28,101)(26,97,29,100)(27,102,30,99)(31,188,34,191)(32,187,35,190)(33,192,36,189)(37,182,40,185)(38,181,41,184)(39,186,42,183)(43,197,46,194)(44,196,47,193)(45,195,48,198)(49,201,52,204)(50,200,53,203)(51,199,54,202)(55,149,58,146)(56,148,59,145)(57,147,60,150)(61,158,64,161)(62,157,65,160)(63,162,66,159)(67,152,70,155)(68,151,71,154)(69,156,72,153)(73,167,76,164)(74,166,77,163)(75,165,78,168)(79,171,82,174)(80,170,83,173)(81,169,84,172)(85,239,88,236)(86,238,89,235)(87,237,90,240)(115,209,118,206)(116,208,119,205)(117,207,120,210)(121,213,124,216)(122,212,125,215)(123,211,126,214)(127,217,130,220)(128,222,131,219)(129,221,132,218)(133,227,136,224)(134,226,137,223)(135,225,138,228)(139,231,142,234)(140,230,143,233)(141,229,144,232), (1,36,39,53,43)(2,31,40,54,44)(3,32,41,49,45)(4,33,42,50,46)(5,34,37,51,47)(6,35,38,52,48)(7,17,237,19,26)(8,18,238,20,27)(9,13,239,21,28)(10,14,240,22,29)(11,15,235,23,30)(12,16,236,24,25)(55,65,68,82,78)(56,66,69,83,73)(57,61,70,84,74)(58,62,71,79,75)(59,63,72,80,76)(60,64,67,81,77)(85,95,98,112,108)(86,96,99,113,103)(87,91,100,114,104)(88,92,101,109,105)(89,93,102,110,106)(90,94,97,111,107)(115,130,123,142,138)(116,131,124,143,133)(117,132,125,144,134)(118,127,126,139,135)(119,128,121,140,136)(120,129,122,141,137)(145,162,153,170,164)(146,157,154,171,165)(147,158,155,172,166)(148,159,156,173,167)(149,160,151,174,168)(150,161,152,169,163)(175,192,183,200,194)(176,187,184,201,195)(177,188,185,202,196)(178,189,186,203,197)(179,190,181,204,198)(180,191,182,199,193)(205,222,213,230,224)(206,217,214,231,225)(207,218,215,232,226)(208,219,216,233,227)(209,220,211,234,228)(210,221,212,229,223), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,221)(8,222)(9,217)(10,218)(11,219)(12,220)(13,206)(14,207)(15,208)(16,209)(17,210)(18,205)(19,229)(20,230)(21,231)(22,232)(23,233)(24,234)(25,211)(26,212)(27,213)(28,214)(29,215)(30,216)(31,54)(32,49)(33,50)(34,51)(35,52)(36,53)(55,78)(56,73)(57,74)(58,75)(59,76)(60,77)(61,84)(62,79)(63,80)(64,81)(65,82)(66,83)(85,105)(86,106)(87,107)(88,108)(89,103)(90,104)(91,111)(92,112)(93,113)(94,114)(95,109)(96,110)(97,100)(98,101)(99,102)(115,135)(116,136)(117,137)(118,138)(119,133)(120,134)(121,124)(122,125)(123,126)(127,142)(128,143)(129,144)(130,139)(131,140)(132,141)(145,194)(146,195)(147,196)(148,197)(149,198)(150,193)(151,181)(152,182)(153,183)(154,184)(155,185)(156,186)(157,201)(158,202)(159,203)(160,204)(161,199)(162,200)(163,180)(164,175)(165,176)(166,177)(167,178)(168,179)(169,191)(170,192)(171,187)(172,188)(173,189)(174,190)(223,237)(224,238)(225,239)(226,240)(227,235)(228,236) );
G=PermutationGroup([[(1,119,59,86),(2,120,60,87),(3,115,55,88),(4,116,56,89),(5,117,57,90),(6,118,58,85),(7,202,229,169),(8,203,230,170),(9,204,231,171),(10,199,232,172),(11,200,233,173),(12,201,234,174),(13,198,225,165),(14,193,226,166),(15,194,227,167),(16,195,228,168),(17,196,223,163),(18,197,224,164),(19,188,221,161),(20,189,222,162),(21,190,217,157),(22,191,218,158),(23,192,219,159),(24,187,220,160),(25,184,211,151),(26,185,212,152),(27,186,213,153),(28,181,214,154),(29,182,215,155),(30,183,216,156),(31,129,64,91),(32,130,65,92),(33,131,66,93),(34,132,61,94),(35,127,62,95),(36,128,63,96),(37,125,70,97),(38,126,71,98),(39,121,72,99),(40,122,67,100),(41,123,68,101),(42,124,69,102),(43,136,76,103),(44,137,77,104),(45,138,78,105),(46,133,73,106),(47,134,74,107),(48,135,75,108),(49,142,82,109),(50,143,83,110),(51,144,84,111),(52,139,79,112),(53,140,80,113),(54,141,81,114),(145,238,178,205),(146,239,179,206),(147,240,180,207),(148,235,175,208),(149,236,176,209),(150,237,177,210)], [(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)], [(1,178,4,175),(2,177,5,180),(3,176,6,179),(7,111,10,114),(8,110,11,113),(9,109,12,112),(13,105,16,108),(14,104,17,107),(15,103,18,106),(19,94,22,91),(20,93,23,96),(21,92,24,95),(25,98,28,101),(26,97,29,100),(27,102,30,99),(31,188,34,191),(32,187,35,190),(33,192,36,189),(37,182,40,185),(38,181,41,184),(39,186,42,183),(43,197,46,194),(44,196,47,193),(45,195,48,198),(49,201,52,204),(50,200,53,203),(51,199,54,202),(55,149,58,146),(56,148,59,145),(57,147,60,150),(61,158,64,161),(62,157,65,160),(63,162,66,159),(67,152,70,155),(68,151,71,154),(69,156,72,153),(73,167,76,164),(74,166,77,163),(75,165,78,168),(79,171,82,174),(80,170,83,173),(81,169,84,172),(85,239,88,236),(86,238,89,235),(87,237,90,240),(115,209,118,206),(116,208,119,205),(117,207,120,210),(121,213,124,216),(122,212,125,215),(123,211,126,214),(127,217,130,220),(128,222,131,219),(129,221,132,218),(133,227,136,224),(134,226,137,223),(135,225,138,228),(139,231,142,234),(140,230,143,233),(141,229,144,232)], [(1,36,39,53,43),(2,31,40,54,44),(3,32,41,49,45),(4,33,42,50,46),(5,34,37,51,47),(6,35,38,52,48),(7,17,237,19,26),(8,18,238,20,27),(9,13,239,21,28),(10,14,240,22,29),(11,15,235,23,30),(12,16,236,24,25),(55,65,68,82,78),(56,66,69,83,73),(57,61,70,84,74),(58,62,71,79,75),(59,63,72,80,76),(60,64,67,81,77),(85,95,98,112,108),(86,96,99,113,103),(87,91,100,114,104),(88,92,101,109,105),(89,93,102,110,106),(90,94,97,111,107),(115,130,123,142,138),(116,131,124,143,133),(117,132,125,144,134),(118,127,126,139,135),(119,128,121,140,136),(120,129,122,141,137),(145,162,153,170,164),(146,157,154,171,165),(147,158,155,172,166),(148,159,156,173,167),(149,160,151,174,168),(150,161,152,169,163),(175,192,183,200,194),(176,187,184,201,195),(177,188,185,202,196),(178,189,186,203,197),(179,190,181,204,198),(180,191,182,199,193),(205,222,213,230,224),(206,217,214,231,225),(207,218,215,232,226),(208,219,216,233,227),(209,220,211,234,228),(210,221,212,229,223)], [(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,221),(8,222),(9,217),(10,218),(11,219),(12,220),(13,206),(14,207),(15,208),(16,209),(17,210),(18,205),(19,229),(20,230),(21,231),(22,232),(23,233),(24,234),(25,211),(26,212),(27,213),(28,214),(29,215),(30,216),(31,54),(32,49),(33,50),(34,51),(35,52),(36,53),(55,78),(56,73),(57,74),(58,75),(59,76),(60,77),(61,84),(62,79),(63,80),(64,81),(65,82),(66,83),(85,105),(86,106),(87,107),(88,108),(89,103),(90,104),(91,111),(92,112),(93,113),(94,114),(95,109),(96,110),(97,100),(98,101),(99,102),(115,135),(116,136),(117,137),(118,138),(119,133),(120,134),(121,124),(122,125),(123,126),(127,142),(128,143),(129,144),(130,139),(131,140),(132,141),(145,194),(146,195),(147,196),(148,197),(149,198),(150,193),(151,181),(152,182),(153,183),(154,184),(155,185),(156,186),(157,201),(158,202),(159,203),(160,204),(161,199),(162,200),(163,180),(164,175),(165,176),(166,177),(167,178),(168,179),(169,191),(170,192),(171,187),(172,188),(173,189),(174,190),(223,237),(224,238),(225,239),(226,240),(227,235),(228,236)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 20A | ··· | 20H | 20I | ··· | 20X | 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 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 20 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 20 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 20 | 20 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | D6 | C4○D4 | D10 | D10 | C4○D12 | C4○D20 | D4⋊2S3 | S3×D5 | C2×S3×D5 | D20⋊5S3 | D6.D10 | Dic5.D6 |
| kernel | (C4×Dic3)⋊D5 | D10⋊Dic3 | C30.Q8 | Dic15⋊5C4 | C3×D10⋊C4 | Dic3×C20 | C30.4Q8 | D10⋊C4 | C4×Dic3 | C2×Dic5 | C2×C20 | C22×D5 | C30 | C2×Dic3 | C2×C12 | C10 | C6 | C10 | C2×C4 | C22 | C2 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 6 | 4 | 2 | 4 | 24 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of (C4×Dic3)⋊D5 ►in GL4(𝔽61) generated by
| 36 | 7 | 0 | 0 |
| 50 | 25 | 0 | 0 |
| 0 | 0 | 50 | 0 |
| 0 | 0 | 0 | 50 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 13 | 19 |
| 0 | 0 | 0 | 47 |
| 11 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 8 | 3 |
| 0 | 0 | 40 | 53 |
| 18 | 1 | 0 | 0 |
| 42 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 60 | 60 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 24 |
| 0 | 0 | 0 | 60 |
G:=sub<GL(4,GF(61))| [36,50,0,0,7,25,0,0,0,0,50,0,0,0,0,50],[60,0,0,0,0,60,0,0,0,0,13,0,0,0,19,47],[11,0,0,0,0,11,0,0,0,0,8,40,0,0,3,53],[18,42,0,0,1,60,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,60,1,0,0,0,0,1,0,0,0,24,60] >;
(C4×Dic3)⋊D5 in GAP, Magma, Sage, TeX
(C_4\times {\rm Dic}_3)\rtimes D_5 % in TeX
G:=Group("(C4xDic3):D5"); // GroupNames label
G:=SmallGroup(480,439);
// by ID
G=gap.SmallGroup(480,439);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,120,422,142,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^6=d^5=e^2=1,c^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a*b^3,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=a^2*b^3*c,e*d*e=d^-1>;
// generators/relations