direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×Dic5⋊4D4, C5⋊3(D4×C12), C15⋊35(C4×D4), C5⋊D4⋊2C12, D10⋊3(C2×C12), Dic5⋊4(C3×D4), C10.16(C6×D4), C6.170(D4×D5), C22⋊2(D5×C12), D10⋊C4⋊9C6, (C4×Dic5)⋊11C6, (C3×Dic5)⋊19D4, Dic5⋊2(C2×C12), C10.D4⋊9C6, C30.330(C2×D4), C23.18(C6×D5), (C12×Dic5)⋊29C2, (C2×C12).272D10, (C22×Dic5)⋊4C6, (C22×C6).74D10, C30.229(C4○D4), C30.178(C22×C4), C10.20(C22×C12), (C2×C60).394C22, (C2×C30).339C23, C6.109(D4⋊2D5), (C22×C30).97C22, (C6×Dic5).235C22, (C2×C4×D5)⋊9C6, C2.2(C3×D4×D5), (C2×C6)⋊7(C4×D5), C2.9(D5×C2×C12), (D5×C2×C12)⋊25C2, (C3×C5⋊D4)⋊5C4, C6.103(C2×C4×D5), (C2×C10)⋊8(C2×C12), (C2×C30)⋊27(C2×C4), (C6×D5)⋊23(C2×C4), (C5×C22⋊C4)⋊9C6, C22⋊C4⋊7(C3×D5), (C2×C6×Dic5)⋊12C2, (C2×C4).23(C6×D5), (C6×C5⋊D4).9C2, (C2×C5⋊D4).2C6, C22.14(D5×C2×C6), (C2×C20).49(C2×C6), (C3×C22⋊C4)⋊15D5, C10.21(C3×C4○D4), C2.2(C3×D4⋊2D5), (C15×C22⋊C4)⋊23C2, (C3×Dic5)⋊17(C2×C4), (C3×D10⋊C4)⋊26C2, (D5×C2×C6).125C22, (C3×C10.D4)⋊25C2, (C22×C10).16(C2×C6), (C2×C10).22(C22×C6), (C2×Dic5).59(C2×C6), (C22×D5).22(C2×C6), (C2×C6).335(C22×D5), SmallGroup(480,674)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×Dic5⋊4D4
G = < a,b,c,d,e | a3=b10=d4=e2=1, c2=b5, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=b-1, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 560 in 188 conjugacy classes, 86 normal (58 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C12, C2×C6, C2×C6, C2×C6, C15, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C3×D5, C30, C30, C4×D4, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, C4×C12, C3×C22⋊C4, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C3×Dic5, C3×Dic5, C60, C6×D5, C6×D5, C2×C30, C2×C30, C2×C30, C4×Dic5, C10.D4, D10⋊C4, C5×C22⋊C4, C2×C4×D5, C22×Dic5, C2×C5⋊D4, D4×C12, D5×C12, C6×Dic5, C6×Dic5, C3×C5⋊D4, C2×C60, D5×C2×C6, C22×C30, Dic5⋊4D4, C12×Dic5, C3×C10.D4, C3×D10⋊C4, C15×C22⋊C4, D5×C2×C12, C2×C6×Dic5, C6×C5⋊D4, C3×Dic5⋊4D4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C23, D5, C12, C2×C6, C22×C4, C2×D4, C4○D4, D10, C2×C12, C3×D4, C22×C6, C3×D5, C4×D4, C4×D5, C22×D5, C22×C12, C6×D4, C3×C4○D4, C6×D5, C2×C4×D5, D4×D5, D4⋊2D5, D4×C12, D5×C12, D5×C2×C6, Dic5⋊4D4, D5×C2×C12, C3×D4×D5, C3×D4⋊2D5, C3×Dic5⋊4D4
►On 240 points
Generators in S240
(1 60 40)(2 51 31)(3 52 32)(4 53 33)(5 54 34)(6 55 35)(7 56 36)(8 57 37)(9 58 38)(10 59 39)(11 225 205)(12 226 206)(13 227 207)(14 228 208)(15 229 209)(16 230 210)(17 221 201)(18 222 202)(19 223 203)(20 224 204)(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)(81 121 101)(82 122 102)(83 123 103)(84 124 104)(85 125 105)(86 126 106)(87 127 107)(88 128 108)(89 129 109)(90 130 110)(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 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 75 6 80)(2 74 7 79)(3 73 8 78)(4 72 9 77)(5 71 10 76)(11 189 16 184)(12 188 17 183)(13 187 18 182)(14 186 19 181)(15 185 20 190)(21 88 26 83)(22 87 27 82)(23 86 28 81)(24 85 29 90)(25 84 30 89)(31 94 36 99)(32 93 37 98)(33 92 38 97)(34 91 39 96)(35 100 40 95)(41 108 46 103)(42 107 47 102)(43 106 48 101)(44 105 49 110)(45 104 50 109)(51 114 56 119)(52 113 57 118)(53 112 58 117)(54 111 59 116)(55 120 60 115)(61 128 66 123)(62 127 67 122)(63 126 68 121)(64 125 69 130)(65 124 70 129)(131 194 136 199)(132 193 137 198)(133 192 138 197)(134 191 139 196)(135 200 140 195)(141 208 146 203)(142 207 147 202)(143 206 148 201)(144 205 149 210)(145 204 150 209)(151 214 156 219)(152 213 157 218)(153 212 158 217)(154 211 159 216)(155 220 160 215)(161 228 166 223)(162 227 167 222)(163 226 168 221)(164 225 169 230)(165 224 170 229)(171 234 176 239)(172 233 177 238)(173 232 178 237)(174 231 179 236)(175 240 180 235)
(1 147 27 140)(2 146 28 139)(3 145 29 138)(4 144 30 137)(5 143 21 136)(6 142 22 135)(7 141 23 134)(8 150 24 133)(9 149 25 132)(10 148 26 131)(11 129 238 112)(12 128 239 111)(13 127 240 120)(14 126 231 119)(15 125 232 118)(16 124 233 117)(17 123 234 116)(18 122 235 115)(19 121 236 114)(20 130 237 113)(31 166 48 159)(32 165 49 158)(33 164 50 157)(34 163 41 156)(35 162 42 155)(36 161 43 154)(37 170 44 153)(38 169 45 152)(39 168 46 151)(40 167 47 160)(51 186 68 179)(52 185 69 178)(53 184 70 177)(54 183 61 176)(55 182 62 175)(56 181 63 174)(57 190 64 173)(58 189 65 172)(59 188 66 171)(60 187 67 180)(71 206 88 199)(72 205 89 198)(73 204 90 197)(74 203 81 196)(75 202 82 195)(76 201 83 194)(77 210 84 193)(78 209 85 192)(79 208 86 191)(80 207 87 200)(91 226 108 219)(92 225 109 218)(93 224 110 217)(94 223 101 216)(95 222 102 215)(96 221 103 214)(97 230 104 213)(98 229 105 212)(99 228 106 211)(100 227 107 220)
(1 140)(2 131)(3 132)(4 133)(5 134)(6 135)(7 136)(8 137)(9 138)(10 139)(11 125)(12 126)(13 127)(14 128)(15 129)(16 130)(17 121)(18 122)(19 123)(20 124)(21 141)(22 142)(23 143)(24 144)(25 145)(26 146)(27 147)(28 148)(29 149)(30 150)(31 151)(32 152)(33 153)(34 154)(35 155)(36 156)(37 157)(38 158)(39 159)(40 160)(41 161)(42 162)(43 163)(44 164)(45 165)(46 166)(47 167)(48 168)(49 169)(50 170)(51 171)(52 172)(53 173)(54 174)(55 175)(56 176)(57 177)(58 178)(59 179)(60 180)(61 181)(62 182)(63 183)(64 184)(65 185)(66 186)(67 187)(68 188)(69 189)(70 190)(71 191)(72 192)(73 193)(74 194)(75 195)(76 196)(77 197)(78 198)(79 199)(80 200)(81 201)(82 202)(83 203)(84 204)(85 205)(86 206)(87 207)(88 208)(89 209)(90 210)(91 211)(92 212)(93 213)(94 214)(95 215)(96 216)(97 217)(98 218)(99 219)(100 220)(101 221)(102 222)(103 223)(104 224)(105 225)(106 226)(107 227)(108 228)(109 229)(110 230)(111 231)(112 232)(113 233)(114 234)(115 235)(116 236)(117 237)(118 238)(119 239)(120 240)
G:=sub<Sym(240)| (1,60,40)(2,51,31)(3,52,32)(4,53,33)(5,54,34)(6,55,35)(7,56,36)(8,57,37)(9,58,38)(10,59,39)(11,225,205)(12,226,206)(13,227,207)(14,228,208)(15,229,209)(16,230,210)(17,221,201)(18,222,202)(19,223,203)(20,224,204)(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)(81,121,101)(82,122,102)(83,123,103)(84,124,104)(85,125,105)(86,126,106)(87,127,107)(88,128,108)(89,129,109)(90,130,110)(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), (1,2,3,4,5,6,7,8,9,10)(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), (1,75,6,80)(2,74,7,79)(3,73,8,78)(4,72,9,77)(5,71,10,76)(11,189,16,184)(12,188,17,183)(13,187,18,182)(14,186,19,181)(15,185,20,190)(21,88,26,83)(22,87,27,82)(23,86,28,81)(24,85,29,90)(25,84,30,89)(31,94,36,99)(32,93,37,98)(33,92,38,97)(34,91,39,96)(35,100,40,95)(41,108,46,103)(42,107,47,102)(43,106,48,101)(44,105,49,110)(45,104,50,109)(51,114,56,119)(52,113,57,118)(53,112,58,117)(54,111,59,116)(55,120,60,115)(61,128,66,123)(62,127,67,122)(63,126,68,121)(64,125,69,130)(65,124,70,129)(131,194,136,199)(132,193,137,198)(133,192,138,197)(134,191,139,196)(135,200,140,195)(141,208,146,203)(142,207,147,202)(143,206,148,201)(144,205,149,210)(145,204,150,209)(151,214,156,219)(152,213,157,218)(153,212,158,217)(154,211,159,216)(155,220,160,215)(161,228,166,223)(162,227,167,222)(163,226,168,221)(164,225,169,230)(165,224,170,229)(171,234,176,239)(172,233,177,238)(173,232,178,237)(174,231,179,236)(175,240,180,235), (1,147,27,140)(2,146,28,139)(3,145,29,138)(4,144,30,137)(5,143,21,136)(6,142,22,135)(7,141,23,134)(8,150,24,133)(9,149,25,132)(10,148,26,131)(11,129,238,112)(12,128,239,111)(13,127,240,120)(14,126,231,119)(15,125,232,118)(16,124,233,117)(17,123,234,116)(18,122,235,115)(19,121,236,114)(20,130,237,113)(31,166,48,159)(32,165,49,158)(33,164,50,157)(34,163,41,156)(35,162,42,155)(36,161,43,154)(37,170,44,153)(38,169,45,152)(39,168,46,151)(40,167,47,160)(51,186,68,179)(52,185,69,178)(53,184,70,177)(54,183,61,176)(55,182,62,175)(56,181,63,174)(57,190,64,173)(58,189,65,172)(59,188,66,171)(60,187,67,180)(71,206,88,199)(72,205,89,198)(73,204,90,197)(74,203,81,196)(75,202,82,195)(76,201,83,194)(77,210,84,193)(78,209,85,192)(79,208,86,191)(80,207,87,200)(91,226,108,219)(92,225,109,218)(93,224,110,217)(94,223,101,216)(95,222,102,215)(96,221,103,214)(97,230,104,213)(98,229,105,212)(99,228,106,211)(100,227,107,220), (1,140)(2,131)(3,132)(4,133)(5,134)(6,135)(7,136)(8,137)(9,138)(10,139)(11,125)(12,126)(13,127)(14,128)(15,129)(16,130)(17,121)(18,122)(19,123)(20,124)(21,141)(22,142)(23,143)(24,144)(25,145)(26,146)(27,147)(28,148)(29,149)(30,150)(31,151)(32,152)(33,153)(34,154)(35,155)(36,156)(37,157)(38,158)(39,159)(40,160)(41,161)(42,162)(43,163)(44,164)(45,165)(46,166)(47,167)(48,168)(49,169)(50,170)(51,171)(52,172)(53,173)(54,174)(55,175)(56,176)(57,177)(58,178)(59,179)(60,180)(61,181)(62,182)(63,183)(64,184)(65,185)(66,186)(67,187)(68,188)(69,189)(70,190)(71,191)(72,192)(73,193)(74,194)(75,195)(76,196)(77,197)(78,198)(79,199)(80,200)(81,201)(82,202)(83,203)(84,204)(85,205)(86,206)(87,207)(88,208)(89,209)(90,210)(91,211)(92,212)(93,213)(94,214)(95,215)(96,216)(97,217)(98,218)(99,219)(100,220)(101,221)(102,222)(103,223)(104,224)(105,225)(106,226)(107,227)(108,228)(109,229)(110,230)(111,231)(112,232)(113,233)(114,234)(115,235)(116,236)(117,237)(118,238)(119,239)(120,240)>;
G:=Group( (1,60,40)(2,51,31)(3,52,32)(4,53,33)(5,54,34)(6,55,35)(7,56,36)(8,57,37)(9,58,38)(10,59,39)(11,225,205)(12,226,206)(13,227,207)(14,228,208)(15,229,209)(16,230,210)(17,221,201)(18,222,202)(19,223,203)(20,224,204)(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)(81,121,101)(82,122,102)(83,123,103)(84,124,104)(85,125,105)(86,126,106)(87,127,107)(88,128,108)(89,129,109)(90,130,110)(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), (1,2,3,4,5,6,7,8,9,10)(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), (1,75,6,80)(2,74,7,79)(3,73,8,78)(4,72,9,77)(5,71,10,76)(11,189,16,184)(12,188,17,183)(13,187,18,182)(14,186,19,181)(15,185,20,190)(21,88,26,83)(22,87,27,82)(23,86,28,81)(24,85,29,90)(25,84,30,89)(31,94,36,99)(32,93,37,98)(33,92,38,97)(34,91,39,96)(35,100,40,95)(41,108,46,103)(42,107,47,102)(43,106,48,101)(44,105,49,110)(45,104,50,109)(51,114,56,119)(52,113,57,118)(53,112,58,117)(54,111,59,116)(55,120,60,115)(61,128,66,123)(62,127,67,122)(63,126,68,121)(64,125,69,130)(65,124,70,129)(131,194,136,199)(132,193,137,198)(133,192,138,197)(134,191,139,196)(135,200,140,195)(141,208,146,203)(142,207,147,202)(143,206,148,201)(144,205,149,210)(145,204,150,209)(151,214,156,219)(152,213,157,218)(153,212,158,217)(154,211,159,216)(155,220,160,215)(161,228,166,223)(162,227,167,222)(163,226,168,221)(164,225,169,230)(165,224,170,229)(171,234,176,239)(172,233,177,238)(173,232,178,237)(174,231,179,236)(175,240,180,235), (1,147,27,140)(2,146,28,139)(3,145,29,138)(4,144,30,137)(5,143,21,136)(6,142,22,135)(7,141,23,134)(8,150,24,133)(9,149,25,132)(10,148,26,131)(11,129,238,112)(12,128,239,111)(13,127,240,120)(14,126,231,119)(15,125,232,118)(16,124,233,117)(17,123,234,116)(18,122,235,115)(19,121,236,114)(20,130,237,113)(31,166,48,159)(32,165,49,158)(33,164,50,157)(34,163,41,156)(35,162,42,155)(36,161,43,154)(37,170,44,153)(38,169,45,152)(39,168,46,151)(40,167,47,160)(51,186,68,179)(52,185,69,178)(53,184,70,177)(54,183,61,176)(55,182,62,175)(56,181,63,174)(57,190,64,173)(58,189,65,172)(59,188,66,171)(60,187,67,180)(71,206,88,199)(72,205,89,198)(73,204,90,197)(74,203,81,196)(75,202,82,195)(76,201,83,194)(77,210,84,193)(78,209,85,192)(79,208,86,191)(80,207,87,200)(91,226,108,219)(92,225,109,218)(93,224,110,217)(94,223,101,216)(95,222,102,215)(96,221,103,214)(97,230,104,213)(98,229,105,212)(99,228,106,211)(100,227,107,220), (1,140)(2,131)(3,132)(4,133)(5,134)(6,135)(7,136)(8,137)(9,138)(10,139)(11,125)(12,126)(13,127)(14,128)(15,129)(16,130)(17,121)(18,122)(19,123)(20,124)(21,141)(22,142)(23,143)(24,144)(25,145)(26,146)(27,147)(28,148)(29,149)(30,150)(31,151)(32,152)(33,153)(34,154)(35,155)(36,156)(37,157)(38,158)(39,159)(40,160)(41,161)(42,162)(43,163)(44,164)(45,165)(46,166)(47,167)(48,168)(49,169)(50,170)(51,171)(52,172)(53,173)(54,174)(55,175)(56,176)(57,177)(58,178)(59,179)(60,180)(61,181)(62,182)(63,183)(64,184)(65,185)(66,186)(67,187)(68,188)(69,189)(70,190)(71,191)(72,192)(73,193)(74,194)(75,195)(76,196)(77,197)(78,198)(79,199)(80,200)(81,201)(82,202)(83,203)(84,204)(85,205)(86,206)(87,207)(88,208)(89,209)(90,210)(91,211)(92,212)(93,213)(94,214)(95,215)(96,216)(97,217)(98,218)(99,219)(100,220)(101,221)(102,222)(103,223)(104,224)(105,225)(106,226)(107,227)(108,228)(109,229)(110,230)(111,231)(112,232)(113,233)(114,234)(115,235)(116,236)(117,237)(118,238)(119,239)(120,240) );
G=PermutationGroup([[(1,60,40),(2,51,31),(3,52,32),(4,53,33),(5,54,34),(6,55,35),(7,56,36),(8,57,37),(9,58,38),(10,59,39),(11,225,205),(12,226,206),(13,227,207),(14,228,208),(15,229,209),(16,230,210),(17,221,201),(18,222,202),(19,223,203),(20,224,204),(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),(81,121,101),(82,122,102),(83,123,103),(84,124,104),(85,125,105),(86,126,106),(87,127,107),(88,128,108),(89,129,109),(90,130,110),(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)], [(1,2,3,4,5,6,7,8,9,10),(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)], [(1,75,6,80),(2,74,7,79),(3,73,8,78),(4,72,9,77),(5,71,10,76),(11,189,16,184),(12,188,17,183),(13,187,18,182),(14,186,19,181),(15,185,20,190),(21,88,26,83),(22,87,27,82),(23,86,28,81),(24,85,29,90),(25,84,30,89),(31,94,36,99),(32,93,37,98),(33,92,38,97),(34,91,39,96),(35,100,40,95),(41,108,46,103),(42,107,47,102),(43,106,48,101),(44,105,49,110),(45,104,50,109),(51,114,56,119),(52,113,57,118),(53,112,58,117),(54,111,59,116),(55,120,60,115),(61,128,66,123),(62,127,67,122),(63,126,68,121),(64,125,69,130),(65,124,70,129),(131,194,136,199),(132,193,137,198),(133,192,138,197),(134,191,139,196),(135,200,140,195),(141,208,146,203),(142,207,147,202),(143,206,148,201),(144,205,149,210),(145,204,150,209),(151,214,156,219),(152,213,157,218),(153,212,158,217),(154,211,159,216),(155,220,160,215),(161,228,166,223),(162,227,167,222),(163,226,168,221),(164,225,169,230),(165,224,170,229),(171,234,176,239),(172,233,177,238),(173,232,178,237),(174,231,179,236),(175,240,180,235)], [(1,147,27,140),(2,146,28,139),(3,145,29,138),(4,144,30,137),(5,143,21,136),(6,142,22,135),(7,141,23,134),(8,150,24,133),(9,149,25,132),(10,148,26,131),(11,129,238,112),(12,128,239,111),(13,127,240,120),(14,126,231,119),(15,125,232,118),(16,124,233,117),(17,123,234,116),(18,122,235,115),(19,121,236,114),(20,130,237,113),(31,166,48,159),(32,165,49,158),(33,164,50,157),(34,163,41,156),(35,162,42,155),(36,161,43,154),(37,170,44,153),(38,169,45,152),(39,168,46,151),(40,167,47,160),(51,186,68,179),(52,185,69,178),(53,184,70,177),(54,183,61,176),(55,182,62,175),(56,181,63,174),(57,190,64,173),(58,189,65,172),(59,188,66,171),(60,187,67,180),(71,206,88,199),(72,205,89,198),(73,204,90,197),(74,203,81,196),(75,202,82,195),(76,201,83,194),(77,210,84,193),(78,209,85,192),(79,208,86,191),(80,207,87,200),(91,226,108,219),(92,225,109,218),(93,224,110,217),(94,223,101,216),(95,222,102,215),(96,221,103,214),(97,230,104,213),(98,229,105,212),(99,228,106,211),(100,227,107,220)], [(1,140),(2,131),(3,132),(4,133),(5,134),(6,135),(7,136),(8,137),(9,138),(10,139),(11,125),(12,126),(13,127),(14,128),(15,129),(16,130),(17,121),(18,122),(19,123),(20,124),(21,141),(22,142),(23,143),(24,144),(25,145),(26,146),(27,147),(28,148),(29,149),(30,150),(31,151),(32,152),(33,153),(34,154),(35,155),(36,156),(37,157),(38,158),(39,159),(40,160),(41,161),(42,162),(43,163),(44,164),(45,165),(46,166),(47,167),(48,168),(49,169),(50,170),(51,171),(52,172),(53,173),(54,174),(55,175),(56,176),(57,177),(58,178),(59,179),(60,180),(61,181),(62,182),(63,183),(64,184),(65,185),(66,186),(67,187),(68,188),(69,189),(70,190),(71,191),(72,192),(73,193),(74,194),(75,195),(76,196),(77,197),(78,198),(79,199),(80,200),(81,201),(82,202),(83,203),(84,204),(85,205),(86,206),(87,207),(88,208),(89,209),(90,210),(91,211),(92,212),(93,213),(94,214),(95,215),(96,216),(97,217),(98,218),(99,219),(100,220),(101,221),(102,222),(103,223),(104,224),(105,225),(106,226),(107,227),(108,228),(109,229),(110,230),(111,231),(112,232),(113,233),(114,234),(115,235),(116,236),(117,237),(118,238),(119,239),(120,240)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 6N | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | ··· | 12H | 12I | ··· | 12P | 12Q | ··· | 12X | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 30A | ··· | 30L | 30M | ··· | 30T | 60A | ··· | 60P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 1 | 1 | 2 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 5 | ··· | 5 | 10 | ··· | 10 | 2 | 2 | 2 | 2 | 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 | 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 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C6 | C6 | C6 | C12 | D4 | D5 | C4○D4 | D10 | D10 | C3×D4 | C3×D5 | C4×D5 | C3×C4○D4 | C6×D5 | C6×D5 | D5×C12 | D4×D5 | D4⋊2D5 | C3×D4×D5 | C3×D4⋊2D5 |
| kernel | C3×Dic5⋊4D4 | C12×Dic5 | C3×C10.D4 | C3×D10⋊C4 | C15×C22⋊C4 | D5×C2×C12 | C2×C6×Dic5 | C6×C5⋊D4 | Dic5⋊4D4 | C3×C5⋊D4 | C4×Dic5 | C10.D4 | D10⋊C4 | C5×C22⋊C4 | C2×C4×D5 | C22×Dic5 | C2×C5⋊D4 | C5⋊D4 | C3×Dic5 | C3×C22⋊C4 | C30 | C2×C12 | C22×C6 | Dic5 | C22⋊C4 | C2×C6 | C10 | C2×C4 | C23 | C22 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 16 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 4 | 8 | 4 | 16 | 2 | 2 | 4 | 4 |
Matrix representation of C3×Dic5⋊4D4 ►in GL5(𝔽61)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 47 | 0 | 0 | 0 |
| 0 | 0 | 47 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 13 |
| 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 1 | 43 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 11 | 0 | 0 | 0 | 0 |
| 0 | 18 | 43 | 0 | 0 |
| 0 | 1 | 43 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 43 | 18 | 0 | 0 |
| 0 | 60 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 60 | 0 |
| 60 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,47,0,0,0,0,0,47,0,0,0,0,0,13,0,0,0,0,0,13],[60,0,0,0,0,0,0,1,0,0,0,60,43,0,0,0,0,0,1,0,0,0,0,0,1],[11,0,0,0,0,0,18,1,0,0,0,43,43,0,0,0,0,0,60,0,0,0,0,0,60],[1,0,0,0,0,0,43,60,0,0,0,18,18,0,0,0,0,0,0,60,0,0,0,1,0],[60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;
C3×Dic5⋊4D4 in GAP, Magma, Sage, TeX
C_3\times {\rm Dic}_5\rtimes_4D_4 % in TeX
G:=Group("C3xDic5:4D4"); // GroupNames label
G:=SmallGroup(480,674);
// by ID
G=gap.SmallGroup(480,674);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,701,555,142,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^10=d^4=e^2=1,c^2=b^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=d*b*d^-1=b^-1,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations