metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic3⋊C4⋊1D5, (C6×D5).57D4, C6.125(D4×D5), D30⋊4C4⋊2C2, C30.102(C2×D4), (C2×C20).177D6, Dic15⋊5C4⋊2C2, D30⋊3C4⋊23C2, C30.16(C4○D4), C6.15(C4○D20), D10⋊Dic3⋊1C2, (C2×C12).256D10, (C2×C30).38C23, C6.4(Q8⋊2D5), (C2×Dic3).5D10, (C22×D5).79D6, C10.18(C4○D12), C2.8(C12.28D10), D10.26(C3⋊D4), C3⋊6(D10.13D4), (C2×C60).379C22, (C2×Dic5).156D6, C15⋊1(C22.D4), C5⋊1(C23.28D6), C2.8(D6.D10), (C2×Dic15).43C22, (C10×Dic3).22C22, (C6×Dic5).177C22, (C22×D15).18C22, (C2×C4×D5)⋊7S3, (D5×C2×C12)⋊17C2, C2.10(D5×C3⋊D4), (C2×C4).119(S3×D5), (C2×C3⋊D20).1C2, C10.27(C2×C3⋊D4), (D5×C2×C6).92C22, C22.127(C2×S3×D5), (C5×Dic3⋊C4)⋊23C2, (C2×C6).50(C22×D5), (C2×C10).50(C22×S3), SmallGroup(480,424)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic3⋊C4⋊D5
G = < a,b,c,d,e | a6=c4=d5=e2=1, b2=a3, bab-1=a-1, ac=ca, ad=da, ae=ea, cbc-1=a3b, bd=db, ebe=bc2, cd=dc, ce=ec, ede=d-1 >
Subgroups: 892 in 156 conjugacy classes, 48 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, C23, D5, C10, Dic3, C12, D6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, D10, D10, C2×C10, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C3×D5, D15, C30, C22.D4, C4×D5, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, Dic3⋊C4, Dic3⋊C4, D6⋊C4, C6.D4, C2×C3⋊D4, C22×C12, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, C6×D5, D30, C2×C30, C10.D4, D10⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, C23.28D6, C3⋊D20, D5×C12, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, D5×C2×C6, C22×D15, D10.13D4, D10⋊Dic3, D30⋊4C4, Dic15⋊5C4, C5×Dic3⋊C4, D30⋊3C4, C2×C3⋊D20, D5×C2×C12, Dic3⋊C4⋊D5
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C3⋊D4, C22×S3, C22.D4, C22×D5, C4○D12, C2×C3⋊D4, S3×D5, C4○D20, D4×D5, Q8⋊2D5, C23.28D6, C2×S3×D5, D10.13D4, D6.D10, C12.28D10, D5×C3⋊D4, Dic3⋊C4⋊D5
►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)
(1 175 4 178)(2 180 5 177)(3 179 6 176)(7 103 10 106)(8 108 11 105)(9 107 12 104)(13 113 16 110)(14 112 17 109)(15 111 18 114)(19 91 22 94)(20 96 23 93)(21 95 24 92)(25 101 28 98)(26 100 29 97)(27 99 30 102)(31 191 34 188)(32 190 35 187)(33 189 36 192)(37 185 40 182)(38 184 41 181)(39 183 42 186)(43 199 46 202)(44 204 47 201)(45 203 48 200)(49 193 52 196)(50 198 53 195)(51 197 54 194)(55 146 58 149)(56 145 59 148)(57 150 60 147)(61 161 64 158)(62 160 65 157)(63 159 66 162)(67 155 70 152)(68 154 71 151)(69 153 72 156)(73 169 76 172)(74 174 77 171)(75 173 78 170)(79 163 82 166)(80 168 83 165)(81 167 84 164)(85 236 88 239)(86 235 89 238)(87 240 90 237)(115 206 118 209)(116 205 119 208)(117 210 120 207)(121 216 124 213)(122 215 125 212)(123 214 126 211)(127 220 130 217)(128 219 131 222)(129 218 132 221)(133 224 136 227)(134 223 137 226)(135 228 138 225)(139 234 142 231)(140 233 143 230)(141 232 144 229)
(1 116 56 86)(2 117 57 87)(3 118 58 88)(4 119 59 89)(5 120 60 90)(6 115 55 85)(7 199 232 169)(8 200 233 170)(9 201 234 171)(10 202 229 172)(11 203 230 173)(12 204 231 174)(13 195 228 165)(14 196 223 166)(15 197 224 167)(16 198 225 168)(17 193 226 163)(18 194 227 164)(19 191 218 161)(20 192 219 162)(21 187 220 157)(22 188 221 158)(23 189 222 159)(24 190 217 160)(25 181 214 151)(26 182 215 152)(27 183 216 153)(28 184 211 154)(29 185 212 155)(30 186 213 156)(31 132 61 91)(32 127 62 92)(33 128 63 93)(34 129 64 94)(35 130 65 95)(36 131 66 96)(37 122 67 97)(38 123 68 98)(39 124 69 99)(40 125 70 100)(41 126 71 101)(42 121 72 102)(43 144 73 103)(44 139 74 104)(45 140 75 105)(46 141 76 106)(47 142 77 107)(48 143 78 108)(49 134 79 109)(50 135 80 110)(51 136 81 111)(52 137 82 112)(53 138 83 113)(54 133 84 114)(145 238 175 208)(146 239 176 209)(147 240 177 210)(148 235 178 205)(149 236 179 206)(150 237 180 207)
(1 36 44 53 38)(2 31 45 54 39)(3 32 46 49 40)(4 33 47 50 41)(5 34 48 51 42)(6 35 43 52 37)(7 14 29 239 21)(8 15 30 240 22)(9 16 25 235 23)(10 17 26 236 24)(11 18 27 237 19)(12 13 28 238 20)(55 65 73 82 67)(56 66 74 83 68)(57 61 75 84 69)(58 62 76 79 70)(59 63 77 80 71)(60 64 78 81 72)(85 95 103 112 97)(86 96 104 113 98)(87 91 105 114 99)(88 92 106 109 100)(89 93 107 110 101)(90 94 108 111 102)(115 130 144 137 122)(116 131 139 138 123)(117 132 140 133 124)(118 127 141 134 125)(119 128 142 135 126)(120 129 143 136 121)(145 162 174 165 154)(146 157 169 166 155)(147 158 170 167 156)(148 159 171 168 151)(149 160 172 163 152)(150 161 173 164 153)(175 192 204 195 184)(176 187 199 196 185)(177 188 200 197 186)(178 189 201 198 181)(179 190 202 193 182)(180 191 203 194 183)(205 222 234 225 214)(206 217 229 226 215)(207 218 230 227 216)(208 219 231 228 211)(209 220 232 223 212)(210 221 233 224 213)
(1 38)(2 39)(3 40)(4 41)(5 42)(6 37)(7 232)(8 233)(9 234)(10 229)(11 230)(12 231)(13 219)(14 220)(15 221)(16 222)(17 217)(18 218)(19 227)(20 228)(21 223)(22 224)(23 225)(24 226)(25 205)(26 206)(27 207)(28 208)(29 209)(30 210)(31 54)(32 49)(33 50)(34 51)(35 52)(36 53)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)(61 84)(62 79)(63 80)(64 81)(65 82)(66 83)(85 97)(86 98)(87 99)(88 100)(89 101)(90 102)(91 114)(92 109)(93 110)(94 111)(95 112)(96 113)(115 122)(116 123)(117 124)(118 125)(119 126)(120 121)(127 134)(128 135)(129 136)(130 137)(131 138)(132 133)(145 184)(146 185)(147 186)(148 181)(149 182)(150 183)(151 178)(152 179)(153 180)(154 175)(155 176)(156 177)(157 196)(158 197)(159 198)(160 193)(161 194)(162 195)(163 190)(164 191)(165 192)(166 187)(167 188)(168 189)(169 199)(170 200)(171 201)(172 202)(173 203)(174 204)(211 238)(212 239)(213 240)(214 235)(215 236)(216 237)
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), (1,175,4,178)(2,180,5,177)(3,179,6,176)(7,103,10,106)(8,108,11,105)(9,107,12,104)(13,113,16,110)(14,112,17,109)(15,111,18,114)(19,91,22,94)(20,96,23,93)(21,95,24,92)(25,101,28,98)(26,100,29,97)(27,99,30,102)(31,191,34,188)(32,190,35,187)(33,189,36,192)(37,185,40,182)(38,184,41,181)(39,183,42,186)(43,199,46,202)(44,204,47,201)(45,203,48,200)(49,193,52,196)(50,198,53,195)(51,197,54,194)(55,146,58,149)(56,145,59,148)(57,150,60,147)(61,161,64,158)(62,160,65,157)(63,159,66,162)(67,155,70,152)(68,154,71,151)(69,153,72,156)(73,169,76,172)(74,174,77,171)(75,173,78,170)(79,163,82,166)(80,168,83,165)(81,167,84,164)(85,236,88,239)(86,235,89,238)(87,240,90,237)(115,206,118,209)(116,205,119,208)(117,210,120,207)(121,216,124,213)(122,215,125,212)(123,214,126,211)(127,220,130,217)(128,219,131,222)(129,218,132,221)(133,224,136,227)(134,223,137,226)(135,228,138,225)(139,234,142,231)(140,233,143,230)(141,232,144,229), (1,116,56,86)(2,117,57,87)(3,118,58,88)(4,119,59,89)(5,120,60,90)(6,115,55,85)(7,199,232,169)(8,200,233,170)(9,201,234,171)(10,202,229,172)(11,203,230,173)(12,204,231,174)(13,195,228,165)(14,196,223,166)(15,197,224,167)(16,198,225,168)(17,193,226,163)(18,194,227,164)(19,191,218,161)(20,192,219,162)(21,187,220,157)(22,188,221,158)(23,189,222,159)(24,190,217,160)(25,181,214,151)(26,182,215,152)(27,183,216,153)(28,184,211,154)(29,185,212,155)(30,186,213,156)(31,132,61,91)(32,127,62,92)(33,128,63,93)(34,129,64,94)(35,130,65,95)(36,131,66,96)(37,122,67,97)(38,123,68,98)(39,124,69,99)(40,125,70,100)(41,126,71,101)(42,121,72,102)(43,144,73,103)(44,139,74,104)(45,140,75,105)(46,141,76,106)(47,142,77,107)(48,143,78,108)(49,134,79,109)(50,135,80,110)(51,136,81,111)(52,137,82,112)(53,138,83,113)(54,133,84,114)(145,238,175,208)(146,239,176,209)(147,240,177,210)(148,235,178,205)(149,236,179,206)(150,237,180,207), (1,36,44,53,38)(2,31,45,54,39)(3,32,46,49,40)(4,33,47,50,41)(5,34,48,51,42)(6,35,43,52,37)(7,14,29,239,21)(8,15,30,240,22)(9,16,25,235,23)(10,17,26,236,24)(11,18,27,237,19)(12,13,28,238,20)(55,65,73,82,67)(56,66,74,83,68)(57,61,75,84,69)(58,62,76,79,70)(59,63,77,80,71)(60,64,78,81,72)(85,95,103,112,97)(86,96,104,113,98)(87,91,105,114,99)(88,92,106,109,100)(89,93,107,110,101)(90,94,108,111,102)(115,130,144,137,122)(116,131,139,138,123)(117,132,140,133,124)(118,127,141,134,125)(119,128,142,135,126)(120,129,143,136,121)(145,162,174,165,154)(146,157,169,166,155)(147,158,170,167,156)(148,159,171,168,151)(149,160,172,163,152)(150,161,173,164,153)(175,192,204,195,184)(176,187,199,196,185)(177,188,200,197,186)(178,189,201,198,181)(179,190,202,193,182)(180,191,203,194,183)(205,222,234,225,214)(206,217,229,226,215)(207,218,230,227,216)(208,219,231,228,211)(209,220,232,223,212)(210,221,233,224,213), (1,38)(2,39)(3,40)(4,41)(5,42)(6,37)(7,232)(8,233)(9,234)(10,229)(11,230)(12,231)(13,219)(14,220)(15,221)(16,222)(17,217)(18,218)(19,227)(20,228)(21,223)(22,224)(23,225)(24,226)(25,205)(26,206)(27,207)(28,208)(29,209)(30,210)(31,54)(32,49)(33,50)(34,51)(35,52)(36,53)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(61,84)(62,79)(63,80)(64,81)(65,82)(66,83)(85,97)(86,98)(87,99)(88,100)(89,101)(90,102)(91,114)(92,109)(93,110)(94,111)(95,112)(96,113)(115,122)(116,123)(117,124)(118,125)(119,126)(120,121)(127,134)(128,135)(129,136)(130,137)(131,138)(132,133)(145,184)(146,185)(147,186)(148,181)(149,182)(150,183)(151,178)(152,179)(153,180)(154,175)(155,176)(156,177)(157,196)(158,197)(159,198)(160,193)(161,194)(162,195)(163,190)(164,191)(165,192)(166,187)(167,188)(168,189)(169,199)(170,200)(171,201)(172,202)(173,203)(174,204)(211,238)(212,239)(213,240)(214,235)(215,236)(216,237)>;
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), (1,175,4,178)(2,180,5,177)(3,179,6,176)(7,103,10,106)(8,108,11,105)(9,107,12,104)(13,113,16,110)(14,112,17,109)(15,111,18,114)(19,91,22,94)(20,96,23,93)(21,95,24,92)(25,101,28,98)(26,100,29,97)(27,99,30,102)(31,191,34,188)(32,190,35,187)(33,189,36,192)(37,185,40,182)(38,184,41,181)(39,183,42,186)(43,199,46,202)(44,204,47,201)(45,203,48,200)(49,193,52,196)(50,198,53,195)(51,197,54,194)(55,146,58,149)(56,145,59,148)(57,150,60,147)(61,161,64,158)(62,160,65,157)(63,159,66,162)(67,155,70,152)(68,154,71,151)(69,153,72,156)(73,169,76,172)(74,174,77,171)(75,173,78,170)(79,163,82,166)(80,168,83,165)(81,167,84,164)(85,236,88,239)(86,235,89,238)(87,240,90,237)(115,206,118,209)(116,205,119,208)(117,210,120,207)(121,216,124,213)(122,215,125,212)(123,214,126,211)(127,220,130,217)(128,219,131,222)(129,218,132,221)(133,224,136,227)(134,223,137,226)(135,228,138,225)(139,234,142,231)(140,233,143,230)(141,232,144,229), (1,116,56,86)(2,117,57,87)(3,118,58,88)(4,119,59,89)(5,120,60,90)(6,115,55,85)(7,199,232,169)(8,200,233,170)(9,201,234,171)(10,202,229,172)(11,203,230,173)(12,204,231,174)(13,195,228,165)(14,196,223,166)(15,197,224,167)(16,198,225,168)(17,193,226,163)(18,194,227,164)(19,191,218,161)(20,192,219,162)(21,187,220,157)(22,188,221,158)(23,189,222,159)(24,190,217,160)(25,181,214,151)(26,182,215,152)(27,183,216,153)(28,184,211,154)(29,185,212,155)(30,186,213,156)(31,132,61,91)(32,127,62,92)(33,128,63,93)(34,129,64,94)(35,130,65,95)(36,131,66,96)(37,122,67,97)(38,123,68,98)(39,124,69,99)(40,125,70,100)(41,126,71,101)(42,121,72,102)(43,144,73,103)(44,139,74,104)(45,140,75,105)(46,141,76,106)(47,142,77,107)(48,143,78,108)(49,134,79,109)(50,135,80,110)(51,136,81,111)(52,137,82,112)(53,138,83,113)(54,133,84,114)(145,238,175,208)(146,239,176,209)(147,240,177,210)(148,235,178,205)(149,236,179,206)(150,237,180,207), (1,36,44,53,38)(2,31,45,54,39)(3,32,46,49,40)(4,33,47,50,41)(5,34,48,51,42)(6,35,43,52,37)(7,14,29,239,21)(8,15,30,240,22)(9,16,25,235,23)(10,17,26,236,24)(11,18,27,237,19)(12,13,28,238,20)(55,65,73,82,67)(56,66,74,83,68)(57,61,75,84,69)(58,62,76,79,70)(59,63,77,80,71)(60,64,78,81,72)(85,95,103,112,97)(86,96,104,113,98)(87,91,105,114,99)(88,92,106,109,100)(89,93,107,110,101)(90,94,108,111,102)(115,130,144,137,122)(116,131,139,138,123)(117,132,140,133,124)(118,127,141,134,125)(119,128,142,135,126)(120,129,143,136,121)(145,162,174,165,154)(146,157,169,166,155)(147,158,170,167,156)(148,159,171,168,151)(149,160,172,163,152)(150,161,173,164,153)(175,192,204,195,184)(176,187,199,196,185)(177,188,200,197,186)(178,189,201,198,181)(179,190,202,193,182)(180,191,203,194,183)(205,222,234,225,214)(206,217,229,226,215)(207,218,230,227,216)(208,219,231,228,211)(209,220,232,223,212)(210,221,233,224,213), (1,38)(2,39)(3,40)(4,41)(5,42)(6,37)(7,232)(8,233)(9,234)(10,229)(11,230)(12,231)(13,219)(14,220)(15,221)(16,222)(17,217)(18,218)(19,227)(20,228)(21,223)(22,224)(23,225)(24,226)(25,205)(26,206)(27,207)(28,208)(29,209)(30,210)(31,54)(32,49)(33,50)(34,51)(35,52)(36,53)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(61,84)(62,79)(63,80)(64,81)(65,82)(66,83)(85,97)(86,98)(87,99)(88,100)(89,101)(90,102)(91,114)(92,109)(93,110)(94,111)(95,112)(96,113)(115,122)(116,123)(117,124)(118,125)(119,126)(120,121)(127,134)(128,135)(129,136)(130,137)(131,138)(132,133)(145,184)(146,185)(147,186)(148,181)(149,182)(150,183)(151,178)(152,179)(153,180)(154,175)(155,176)(156,177)(157,196)(158,197)(159,198)(160,193)(161,194)(162,195)(163,190)(164,191)(165,192)(166,187)(167,188)(168,189)(169,199)(170,200)(171,201)(172,202)(173,203)(174,204)(211,238)(212,239)(213,240)(214,235)(215,236)(216,237) );
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)], [(1,175,4,178),(2,180,5,177),(3,179,6,176),(7,103,10,106),(8,108,11,105),(9,107,12,104),(13,113,16,110),(14,112,17,109),(15,111,18,114),(19,91,22,94),(20,96,23,93),(21,95,24,92),(25,101,28,98),(26,100,29,97),(27,99,30,102),(31,191,34,188),(32,190,35,187),(33,189,36,192),(37,185,40,182),(38,184,41,181),(39,183,42,186),(43,199,46,202),(44,204,47,201),(45,203,48,200),(49,193,52,196),(50,198,53,195),(51,197,54,194),(55,146,58,149),(56,145,59,148),(57,150,60,147),(61,161,64,158),(62,160,65,157),(63,159,66,162),(67,155,70,152),(68,154,71,151),(69,153,72,156),(73,169,76,172),(74,174,77,171),(75,173,78,170),(79,163,82,166),(80,168,83,165),(81,167,84,164),(85,236,88,239),(86,235,89,238),(87,240,90,237),(115,206,118,209),(116,205,119,208),(117,210,120,207),(121,216,124,213),(122,215,125,212),(123,214,126,211),(127,220,130,217),(128,219,131,222),(129,218,132,221),(133,224,136,227),(134,223,137,226),(135,228,138,225),(139,234,142,231),(140,233,143,230),(141,232,144,229)], [(1,116,56,86),(2,117,57,87),(3,118,58,88),(4,119,59,89),(5,120,60,90),(6,115,55,85),(7,199,232,169),(8,200,233,170),(9,201,234,171),(10,202,229,172),(11,203,230,173),(12,204,231,174),(13,195,228,165),(14,196,223,166),(15,197,224,167),(16,198,225,168),(17,193,226,163),(18,194,227,164),(19,191,218,161),(20,192,219,162),(21,187,220,157),(22,188,221,158),(23,189,222,159),(24,190,217,160),(25,181,214,151),(26,182,215,152),(27,183,216,153),(28,184,211,154),(29,185,212,155),(30,186,213,156),(31,132,61,91),(32,127,62,92),(33,128,63,93),(34,129,64,94),(35,130,65,95),(36,131,66,96),(37,122,67,97),(38,123,68,98),(39,124,69,99),(40,125,70,100),(41,126,71,101),(42,121,72,102),(43,144,73,103),(44,139,74,104),(45,140,75,105),(46,141,76,106),(47,142,77,107),(48,143,78,108),(49,134,79,109),(50,135,80,110),(51,136,81,111),(52,137,82,112),(53,138,83,113),(54,133,84,114),(145,238,175,208),(146,239,176,209),(147,240,177,210),(148,235,178,205),(149,236,179,206),(150,237,180,207)], [(1,36,44,53,38),(2,31,45,54,39),(3,32,46,49,40),(4,33,47,50,41),(5,34,48,51,42),(6,35,43,52,37),(7,14,29,239,21),(8,15,30,240,22),(9,16,25,235,23),(10,17,26,236,24),(11,18,27,237,19),(12,13,28,238,20),(55,65,73,82,67),(56,66,74,83,68),(57,61,75,84,69),(58,62,76,79,70),(59,63,77,80,71),(60,64,78,81,72),(85,95,103,112,97),(86,96,104,113,98),(87,91,105,114,99),(88,92,106,109,100),(89,93,107,110,101),(90,94,108,111,102),(115,130,144,137,122),(116,131,139,138,123),(117,132,140,133,124),(118,127,141,134,125),(119,128,142,135,126),(120,129,143,136,121),(145,162,174,165,154),(146,157,169,166,155),(147,158,170,167,156),(148,159,171,168,151),(149,160,172,163,152),(150,161,173,164,153),(175,192,204,195,184),(176,187,199,196,185),(177,188,200,197,186),(178,189,201,198,181),(179,190,202,193,182),(180,191,203,194,183),(205,222,234,225,214),(206,217,229,226,215),(207,218,230,227,216),(208,219,231,228,211),(209,220,232,223,212),(210,221,233,224,213)], [(1,38),(2,39),(3,40),(4,41),(5,42),(6,37),(7,232),(8,233),(9,234),(10,229),(11,230),(12,231),(13,219),(14,220),(15,221),(16,222),(17,217),(18,218),(19,227),(20,228),(21,223),(22,224),(23,225),(24,226),(25,205),(26,206),(27,207),(28,208),(29,209),(30,210),(31,54),(32,49),(33,50),(34,51),(35,52),(36,53),(55,67),(56,68),(57,69),(58,70),(59,71),(60,72),(61,84),(62,79),(63,80),(64,81),(65,82),(66,83),(85,97),(86,98),(87,99),(88,100),(89,101),(90,102),(91,114),(92,109),(93,110),(94,111),(95,112),(96,113),(115,122),(116,123),(117,124),(118,125),(119,126),(120,121),(127,134),(128,135),(129,136),(130,137),(131,138),(132,133),(145,184),(146,185),(147,186),(148,181),(149,182),(150,183),(151,178),(152,179),(153,180),(154,175),(155,176),(156,177),(157,196),(158,197),(159,198),(160,193),(161,194),(162,195),(163,190),(164,191),(165,192),(166,187),(167,188),(168,189),(169,199),(170,200),(171,201),(172,202),(173,203),(174,204),(211,238),(212,239),(213,240),(214,235),(215,236),(216,237)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 60 | 2 | 2 | 2 | 10 | 10 | 12 | 12 | 60 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 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 | 1 | 2 | 2 | 2 | 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 | C2 | S3 | D4 | D5 | D6 | D6 | D6 | C4○D4 | D10 | D10 | C3⋊D4 | C4○D12 | C4○D20 | S3×D5 | D4×D5 | Q8⋊2D5 | C2×S3×D5 | D6.D10 | C12.28D10 | D5×C3⋊D4 |
| kernel | Dic3⋊C4⋊D5 | D10⋊Dic3 | D30⋊4C4 | Dic15⋊5C4 | C5×Dic3⋊C4 | D30⋊3C4 | C2×C3⋊D20 | D5×C2×C12 | C2×C4×D5 | C6×D5 | Dic3⋊C4 | C2×Dic5 | C2×C20 | C22×D5 | C30 | C2×Dic3 | C2×C12 | D10 | C10 | C6 | C2×C4 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 4 | 2 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of Dic3⋊C4⋊D5 ►in GL4(𝔽61) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 48 | 27 |
| 0 | 0 | 0 | 14 |
| 47 | 16 | 0 | 0 |
| 45 | 14 | 0 | 0 |
| 0 | 0 | 3 | 17 |
| 0 | 0 | 3 | 58 |
| 50 | 0 | 0 | 0 |
| 0 | 50 | 0 | 0 |
| 0 | 0 | 50 | 22 |
| 0 | 0 | 0 | 11 |
| 0 | 1 | 0 | 0 |
| 60 | 17 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 59 |
| 0 | 0 | 0 | 60 |
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,48,0,0,0,27,14],[47,45,0,0,16,14,0,0,0,0,3,3,0,0,17,58],[50,0,0,0,0,50,0,0,0,0,50,0,0,0,22,11],[0,60,0,0,1,17,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,59,60] >;
Dic3⋊C4⋊D5 in GAP, Magma, Sage, TeX
{\rm Dic}_3\rtimes C_4\rtimes D_5 % in TeX
G:=Group("Dic3:C4:D5"); // GroupNames label
G:=SmallGroup(480,424);
// by ID
G=gap.SmallGroup(480,424);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,120,219,58,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=c^4=d^5=e^2=1,b^2=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=a^3*b,b*d=d*b,e*b*e=b*c^2,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations