direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×D4.Dic3, C60.290C23, (C3×D4).C20, (C3×Q8).C20, D4.(C5×Dic3), C15⋊29(C8○D4), (D4×C15).5C4, (Q8×C15).5C4, C60.185(C2×C4), C12.15(C2×C20), (C2×C20).368D6, C4.Dic3⋊8C10, (C5×D4).3Dic3, Q8.2(C5×Dic3), (C5×Q8).6Dic3, C4.5(C10×Dic3), C6.27(C22×C20), C20.55(C2×Dic3), C20.248(C22×S3), C12.42(C22×C10), (C2×C60).369C22, C30.234(C22×C4), C22.1(C10×Dic3), C10.50(C22×Dic3), C3⋊3(C5×C8○D4), (C2×C3⋊C8)⋊7C10, (C10×C3⋊C8)⋊21C2, C4.42(S3×C2×C10), C3⋊C8.13(C2×C10), (C2×C6).7(C2×C20), C4○D4.5(C5×S3), C2.8(Dic3×C2×C10), (C2×C4).58(S3×C10), (C3×C4○D4).2C10, (C5×C4○D4).10S3, (C15×C4○D4).8C2, (C5×C3⋊C8).49C22, (C2×C12).43(C2×C10), (C2×C30).131(C2×C4), (C5×C4.Dic3)⋊20C2, (C2×C10).28(C2×Dic3), SmallGroup(480,827)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×D4.Dic3
G = < a,b,c,d,e | a5=b4=c2=1, d6=b2, e2=b2d3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d5 >
Subgroups: 180 in 124 conjugacy classes, 90 normal (24 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, D4, Q8, C10, C10, C12, C12, C2×C6, C15, C2×C8, M4(2), C4○D4, C20, C20, C2×C10, C3⋊C8, C3⋊C8, C2×C12, C3×D4, C3×Q8, C30, C30, C8○D4, C40, C2×C20, C5×D4, C5×Q8, C2×C3⋊C8, C4.Dic3, C3×C4○D4, C60, C60, C2×C30, C2×C40, C5×M4(2), C5×C4○D4, D4.Dic3, C5×C3⋊C8, C5×C3⋊C8, C2×C60, D4×C15, Q8×C15, C5×C8○D4, C10×C3⋊C8, C5×C4.Dic3, C15×C4○D4, C5×D4.Dic3
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, C23, C10, Dic3, D6, C22×C4, C20, C2×C10, C2×Dic3, C22×S3, C5×S3, C8○D4, C2×C20, C22×C10, C22×Dic3, C5×Dic3, S3×C10, C22×C20, D4.Dic3, C10×Dic3, S3×C2×C10, C5×C8○D4, Dic3×C2×C10, C5×D4.Dic3
►On 240 points
Generators in S240
(1 55 46 25 19)(2 56 47 26 20)(3 57 48 27 21)(4 58 37 28 22)(5 59 38 29 23)(6 60 39 30 24)(7 49 40 31 13)(8 50 41 32 14)(9 51 42 33 15)(10 52 43 34 16)(11 53 44 35 17)(12 54 45 36 18)(61 115 106 94 76)(62 116 107 95 77)(63 117 108 96 78)(64 118 97 85 79)(65 119 98 86 80)(66 120 99 87 81)(67 109 100 88 82)(68 110 101 89 83)(69 111 102 90 84)(70 112 103 91 73)(71 113 104 92 74)(72 114 105 93 75)(121 177 164 155 133)(122 178 165 156 134)(123 179 166 145 135)(124 180 167 146 136)(125 169 168 147 137)(126 170 157 148 138)(127 171 158 149 139)(128 172 159 150 140)(129 173 160 151 141)(130 174 161 152 142)(131 175 162 153 143)(132 176 163 154 144)(181 238 220 212 195)(182 239 221 213 196)(183 240 222 214 197)(184 229 223 215 198)(185 230 224 216 199)(186 231 225 205 200)(187 232 226 206 201)(188 233 227 207 202)(189 234 228 208 203)(190 235 217 209 204)(191 236 218 210 193)(192 237 219 211 194)
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 22 19 16)(14 23 20 17)(15 24 21 18)(25 34 31 28)(26 35 32 29)(27 36 33 30)(37 46 43 40)(38 47 44 41)(39 48 45 42)(49 58 55 52)(50 59 56 53)(51 60 57 54)(61 64 67 70)(62 65 68 71)(63 66 69 72)(73 76 79 82)(74 77 80 83)(75 78 81 84)(85 88 91 94)(86 89 92 95)(87 90 93 96)(97 100 103 106)(98 101 104 107)(99 102 105 108)(109 112 115 118)(110 113 116 119)(111 114 117 120)(121 130 127 124)(122 131 128 125)(123 132 129 126)(133 142 139 136)(134 143 140 137)(135 144 141 138)(145 154 151 148)(146 155 152 149)(147 156 153 150)(157 166 163 160)(158 167 164 161)(159 168 165 162)(169 178 175 172)(170 179 176 173)(171 180 177 174)(181 184 187 190)(182 185 188 191)(183 186 189 192)(193 196 199 202)(194 197 200 203)(195 198 201 204)(205 208 211 214)(206 209 212 215)(207 210 213 216)(217 220 223 226)(218 221 224 227)(219 222 225 228)(229 232 235 238)(230 233 236 239)(231 234 237 240)
(1 64)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 71)(9 72)(10 61)(11 62)(12 63)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 79)(20 80)(21 81)(22 82)(23 83)(24 84)(25 85)(26 86)(27 87)(28 88)(29 89)(30 90)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 100)(38 101)(39 102)(40 103)(41 104)(42 105)(43 106)(44 107)(45 108)(46 97)(47 98)(48 99)(49 112)(50 113)(51 114)(52 115)(53 116)(54 117)(55 118)(56 119)(57 120)(58 109)(59 110)(60 111)(121 183)(122 184)(123 185)(124 186)(125 187)(126 188)(127 189)(128 190)(129 191)(130 192)(131 181)(132 182)(133 197)(134 198)(135 199)(136 200)(137 201)(138 202)(139 203)(140 204)(141 193)(142 194)(143 195)(144 196)(145 216)(146 205)(147 206)(148 207)(149 208)(150 209)(151 210)(152 211)(153 212)(154 213)(155 214)(156 215)(157 227)(158 228)(159 217)(160 218)(161 219)(162 220)(163 221)(164 222)(165 223)(166 224)(167 225)(168 226)(169 232)(170 233)(171 234)(172 235)(173 236)(174 237)(175 238)(176 239)(177 240)(178 229)(179 230)(180 231)
(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)
(1 130 10 127 7 124 4 121)(2 123 11 132 8 129 5 126)(3 128 12 125 9 122 6 131)(13 136 22 133 19 142 16 139)(14 141 23 138 20 135 17 144)(15 134 24 143 21 140 18 137)(25 152 34 149 31 146 28 155)(26 145 35 154 32 151 29 148)(27 150 36 147 33 156 30 153)(37 164 46 161 43 158 40 167)(38 157 47 166 44 163 41 160)(39 162 48 159 45 168 42 165)(49 180 58 177 55 174 52 171)(50 173 59 170 56 179 53 176)(51 178 60 175 57 172 54 169)(61 189 70 186 67 183 64 192)(62 182 71 191 68 188 65 185)(63 187 72 184 69 181 66 190)(73 200 82 197 79 194 76 203)(74 193 83 202 80 199 77 196)(75 198 84 195 81 204 78 201)(85 211 94 208 91 205 88 214)(86 216 95 213 92 210 89 207)(87 209 96 206 93 215 90 212)(97 219 106 228 103 225 100 222)(98 224 107 221 104 218 101 227)(99 217 108 226 105 223 102 220)(109 240 118 237 115 234 112 231)(110 233 119 230 116 239 113 236)(111 238 120 235 117 232 114 229)
G:=sub<Sym(240)| (1,55,46,25,19)(2,56,47,26,20)(3,57,48,27,21)(4,58,37,28,22)(5,59,38,29,23)(6,60,39,30,24)(7,49,40,31,13)(8,50,41,32,14)(9,51,42,33,15)(10,52,43,34,16)(11,53,44,35,17)(12,54,45,36,18)(61,115,106,94,76)(62,116,107,95,77)(63,117,108,96,78)(64,118,97,85,79)(65,119,98,86,80)(66,120,99,87,81)(67,109,100,88,82)(68,110,101,89,83)(69,111,102,90,84)(70,112,103,91,73)(71,113,104,92,74)(72,114,105,93,75)(121,177,164,155,133)(122,178,165,156,134)(123,179,166,145,135)(124,180,167,146,136)(125,169,168,147,137)(126,170,157,148,138)(127,171,158,149,139)(128,172,159,150,140)(129,173,160,151,141)(130,174,161,152,142)(131,175,162,153,143)(132,176,163,154,144)(181,238,220,212,195)(182,239,221,213,196)(183,240,222,214,197)(184,229,223,215,198)(185,230,224,216,199)(186,231,225,205,200)(187,232,226,206,201)(188,233,227,207,202)(189,234,228,208,203)(190,235,217,209,204)(191,236,218,210,193)(192,237,219,211,194), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,22,19,16)(14,23,20,17)(15,24,21,18)(25,34,31,28)(26,35,32,29)(27,36,33,30)(37,46,43,40)(38,47,44,41)(39,48,45,42)(49,58,55,52)(50,59,56,53)(51,60,57,54)(61,64,67,70)(62,65,68,71)(63,66,69,72)(73,76,79,82)(74,77,80,83)(75,78,81,84)(85,88,91,94)(86,89,92,95)(87,90,93,96)(97,100,103,106)(98,101,104,107)(99,102,105,108)(109,112,115,118)(110,113,116,119)(111,114,117,120)(121,130,127,124)(122,131,128,125)(123,132,129,126)(133,142,139,136)(134,143,140,137)(135,144,141,138)(145,154,151,148)(146,155,152,149)(147,156,153,150)(157,166,163,160)(158,167,164,161)(159,168,165,162)(169,178,175,172)(170,179,176,173)(171,180,177,174)(181,184,187,190)(182,185,188,191)(183,186,189,192)(193,196,199,202)(194,197,200,203)(195,198,201,204)(205,208,211,214)(206,209,212,215)(207,210,213,216)(217,220,223,226)(218,221,224,227)(219,222,225,228)(229,232,235,238)(230,233,236,239)(231,234,237,240), (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,72)(10,61)(11,62)(12,63)(13,73)(14,74)(15,75)(16,76)(17,77)(18,78)(19,79)(20,80)(21,81)(22,82)(23,83)(24,84)(25,85)(26,86)(27,87)(28,88)(29,89)(30,90)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,100)(38,101)(39,102)(40,103)(41,104)(42,105)(43,106)(44,107)(45,108)(46,97)(47,98)(48,99)(49,112)(50,113)(51,114)(52,115)(53,116)(54,117)(55,118)(56,119)(57,120)(58,109)(59,110)(60,111)(121,183)(122,184)(123,185)(124,186)(125,187)(126,188)(127,189)(128,190)(129,191)(130,192)(131,181)(132,182)(133,197)(134,198)(135,199)(136,200)(137,201)(138,202)(139,203)(140,204)(141,193)(142,194)(143,195)(144,196)(145,216)(146,205)(147,206)(148,207)(149,208)(150,209)(151,210)(152,211)(153,212)(154,213)(155,214)(156,215)(157,227)(158,228)(159,217)(160,218)(161,219)(162,220)(163,221)(164,222)(165,223)(166,224)(167,225)(168,226)(169,232)(170,233)(171,234)(172,235)(173,236)(174,237)(175,238)(176,239)(177,240)(178,229)(179,230)(180,231), (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), (1,130,10,127,7,124,4,121)(2,123,11,132,8,129,5,126)(3,128,12,125,9,122,6,131)(13,136,22,133,19,142,16,139)(14,141,23,138,20,135,17,144)(15,134,24,143,21,140,18,137)(25,152,34,149,31,146,28,155)(26,145,35,154,32,151,29,148)(27,150,36,147,33,156,30,153)(37,164,46,161,43,158,40,167)(38,157,47,166,44,163,41,160)(39,162,48,159,45,168,42,165)(49,180,58,177,55,174,52,171)(50,173,59,170,56,179,53,176)(51,178,60,175,57,172,54,169)(61,189,70,186,67,183,64,192)(62,182,71,191,68,188,65,185)(63,187,72,184,69,181,66,190)(73,200,82,197,79,194,76,203)(74,193,83,202,80,199,77,196)(75,198,84,195,81,204,78,201)(85,211,94,208,91,205,88,214)(86,216,95,213,92,210,89,207)(87,209,96,206,93,215,90,212)(97,219,106,228,103,225,100,222)(98,224,107,221,104,218,101,227)(99,217,108,226,105,223,102,220)(109,240,118,237,115,234,112,231)(110,233,119,230,116,239,113,236)(111,238,120,235,117,232,114,229)>;
G:=Group( (1,55,46,25,19)(2,56,47,26,20)(3,57,48,27,21)(4,58,37,28,22)(5,59,38,29,23)(6,60,39,30,24)(7,49,40,31,13)(8,50,41,32,14)(9,51,42,33,15)(10,52,43,34,16)(11,53,44,35,17)(12,54,45,36,18)(61,115,106,94,76)(62,116,107,95,77)(63,117,108,96,78)(64,118,97,85,79)(65,119,98,86,80)(66,120,99,87,81)(67,109,100,88,82)(68,110,101,89,83)(69,111,102,90,84)(70,112,103,91,73)(71,113,104,92,74)(72,114,105,93,75)(121,177,164,155,133)(122,178,165,156,134)(123,179,166,145,135)(124,180,167,146,136)(125,169,168,147,137)(126,170,157,148,138)(127,171,158,149,139)(128,172,159,150,140)(129,173,160,151,141)(130,174,161,152,142)(131,175,162,153,143)(132,176,163,154,144)(181,238,220,212,195)(182,239,221,213,196)(183,240,222,214,197)(184,229,223,215,198)(185,230,224,216,199)(186,231,225,205,200)(187,232,226,206,201)(188,233,227,207,202)(189,234,228,208,203)(190,235,217,209,204)(191,236,218,210,193)(192,237,219,211,194), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,22,19,16)(14,23,20,17)(15,24,21,18)(25,34,31,28)(26,35,32,29)(27,36,33,30)(37,46,43,40)(38,47,44,41)(39,48,45,42)(49,58,55,52)(50,59,56,53)(51,60,57,54)(61,64,67,70)(62,65,68,71)(63,66,69,72)(73,76,79,82)(74,77,80,83)(75,78,81,84)(85,88,91,94)(86,89,92,95)(87,90,93,96)(97,100,103,106)(98,101,104,107)(99,102,105,108)(109,112,115,118)(110,113,116,119)(111,114,117,120)(121,130,127,124)(122,131,128,125)(123,132,129,126)(133,142,139,136)(134,143,140,137)(135,144,141,138)(145,154,151,148)(146,155,152,149)(147,156,153,150)(157,166,163,160)(158,167,164,161)(159,168,165,162)(169,178,175,172)(170,179,176,173)(171,180,177,174)(181,184,187,190)(182,185,188,191)(183,186,189,192)(193,196,199,202)(194,197,200,203)(195,198,201,204)(205,208,211,214)(206,209,212,215)(207,210,213,216)(217,220,223,226)(218,221,224,227)(219,222,225,228)(229,232,235,238)(230,233,236,239)(231,234,237,240), (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,72)(10,61)(11,62)(12,63)(13,73)(14,74)(15,75)(16,76)(17,77)(18,78)(19,79)(20,80)(21,81)(22,82)(23,83)(24,84)(25,85)(26,86)(27,87)(28,88)(29,89)(30,90)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,100)(38,101)(39,102)(40,103)(41,104)(42,105)(43,106)(44,107)(45,108)(46,97)(47,98)(48,99)(49,112)(50,113)(51,114)(52,115)(53,116)(54,117)(55,118)(56,119)(57,120)(58,109)(59,110)(60,111)(121,183)(122,184)(123,185)(124,186)(125,187)(126,188)(127,189)(128,190)(129,191)(130,192)(131,181)(132,182)(133,197)(134,198)(135,199)(136,200)(137,201)(138,202)(139,203)(140,204)(141,193)(142,194)(143,195)(144,196)(145,216)(146,205)(147,206)(148,207)(149,208)(150,209)(151,210)(152,211)(153,212)(154,213)(155,214)(156,215)(157,227)(158,228)(159,217)(160,218)(161,219)(162,220)(163,221)(164,222)(165,223)(166,224)(167,225)(168,226)(169,232)(170,233)(171,234)(172,235)(173,236)(174,237)(175,238)(176,239)(177,240)(178,229)(179,230)(180,231), (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), (1,130,10,127,7,124,4,121)(2,123,11,132,8,129,5,126)(3,128,12,125,9,122,6,131)(13,136,22,133,19,142,16,139)(14,141,23,138,20,135,17,144)(15,134,24,143,21,140,18,137)(25,152,34,149,31,146,28,155)(26,145,35,154,32,151,29,148)(27,150,36,147,33,156,30,153)(37,164,46,161,43,158,40,167)(38,157,47,166,44,163,41,160)(39,162,48,159,45,168,42,165)(49,180,58,177,55,174,52,171)(50,173,59,170,56,179,53,176)(51,178,60,175,57,172,54,169)(61,189,70,186,67,183,64,192)(62,182,71,191,68,188,65,185)(63,187,72,184,69,181,66,190)(73,200,82,197,79,194,76,203)(74,193,83,202,80,199,77,196)(75,198,84,195,81,204,78,201)(85,211,94,208,91,205,88,214)(86,216,95,213,92,210,89,207)(87,209,96,206,93,215,90,212)(97,219,106,228,103,225,100,222)(98,224,107,221,104,218,101,227)(99,217,108,226,105,223,102,220)(109,240,118,237,115,234,112,231)(110,233,119,230,116,239,113,236)(111,238,120,235,117,232,114,229) );
G=PermutationGroup([[(1,55,46,25,19),(2,56,47,26,20),(3,57,48,27,21),(4,58,37,28,22),(5,59,38,29,23),(6,60,39,30,24),(7,49,40,31,13),(8,50,41,32,14),(9,51,42,33,15),(10,52,43,34,16),(11,53,44,35,17),(12,54,45,36,18),(61,115,106,94,76),(62,116,107,95,77),(63,117,108,96,78),(64,118,97,85,79),(65,119,98,86,80),(66,120,99,87,81),(67,109,100,88,82),(68,110,101,89,83),(69,111,102,90,84),(70,112,103,91,73),(71,113,104,92,74),(72,114,105,93,75),(121,177,164,155,133),(122,178,165,156,134),(123,179,166,145,135),(124,180,167,146,136),(125,169,168,147,137),(126,170,157,148,138),(127,171,158,149,139),(128,172,159,150,140),(129,173,160,151,141),(130,174,161,152,142),(131,175,162,153,143),(132,176,163,154,144),(181,238,220,212,195),(182,239,221,213,196),(183,240,222,214,197),(184,229,223,215,198),(185,230,224,216,199),(186,231,225,205,200),(187,232,226,206,201),(188,233,227,207,202),(189,234,228,208,203),(190,235,217,209,204),(191,236,218,210,193),(192,237,219,211,194)], [(1,10,7,4),(2,11,8,5),(3,12,9,6),(13,22,19,16),(14,23,20,17),(15,24,21,18),(25,34,31,28),(26,35,32,29),(27,36,33,30),(37,46,43,40),(38,47,44,41),(39,48,45,42),(49,58,55,52),(50,59,56,53),(51,60,57,54),(61,64,67,70),(62,65,68,71),(63,66,69,72),(73,76,79,82),(74,77,80,83),(75,78,81,84),(85,88,91,94),(86,89,92,95),(87,90,93,96),(97,100,103,106),(98,101,104,107),(99,102,105,108),(109,112,115,118),(110,113,116,119),(111,114,117,120),(121,130,127,124),(122,131,128,125),(123,132,129,126),(133,142,139,136),(134,143,140,137),(135,144,141,138),(145,154,151,148),(146,155,152,149),(147,156,153,150),(157,166,163,160),(158,167,164,161),(159,168,165,162),(169,178,175,172),(170,179,176,173),(171,180,177,174),(181,184,187,190),(182,185,188,191),(183,186,189,192),(193,196,199,202),(194,197,200,203),(195,198,201,204),(205,208,211,214),(206,209,212,215),(207,210,213,216),(217,220,223,226),(218,221,224,227),(219,222,225,228),(229,232,235,238),(230,233,236,239),(231,234,237,240)], [(1,64),(2,65),(3,66),(4,67),(5,68),(6,69),(7,70),(8,71),(9,72),(10,61),(11,62),(12,63),(13,73),(14,74),(15,75),(16,76),(17,77),(18,78),(19,79),(20,80),(21,81),(22,82),(23,83),(24,84),(25,85),(26,86),(27,87),(28,88),(29,89),(30,90),(31,91),(32,92),(33,93),(34,94),(35,95),(36,96),(37,100),(38,101),(39,102),(40,103),(41,104),(42,105),(43,106),(44,107),(45,108),(46,97),(47,98),(48,99),(49,112),(50,113),(51,114),(52,115),(53,116),(54,117),(55,118),(56,119),(57,120),(58,109),(59,110),(60,111),(121,183),(122,184),(123,185),(124,186),(125,187),(126,188),(127,189),(128,190),(129,191),(130,192),(131,181),(132,182),(133,197),(134,198),(135,199),(136,200),(137,201),(138,202),(139,203),(140,204),(141,193),(142,194),(143,195),(144,196),(145,216),(146,205),(147,206),(148,207),(149,208),(150,209),(151,210),(152,211),(153,212),(154,213),(155,214),(156,215),(157,227),(158,228),(159,217),(160,218),(161,219),(162,220),(163,221),(164,222),(165,223),(166,224),(167,225),(168,226),(169,232),(170,233),(171,234),(172,235),(173,236),(174,237),(175,238),(176,239),(177,240),(178,229),(179,230),(180,231)], [(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)], [(1,130,10,127,7,124,4,121),(2,123,11,132,8,129,5,126),(3,128,12,125,9,122,6,131),(13,136,22,133,19,142,16,139),(14,141,23,138,20,135,17,144),(15,134,24,143,21,140,18,137),(25,152,34,149,31,146,28,155),(26,145,35,154,32,151,29,148),(27,150,36,147,33,156,30,153),(37,164,46,161,43,158,40,167),(38,157,47,166,44,163,41,160),(39,162,48,159,45,168,42,165),(49,180,58,177,55,174,52,171),(50,173,59,170,56,179,53,176),(51,178,60,175,57,172,54,169),(61,189,70,186,67,183,64,192),(62,182,71,191,68,188,65,185),(63,187,72,184,69,181,66,190),(73,200,82,197,79,194,76,203),(74,193,83,202,80,199,77,196),(75,198,84,195,81,204,78,201),(85,211,94,208,91,205,88,214),(86,216,95,213,92,210,89,207),(87,209,96,206,93,215,90,212),(97,219,106,228,103,225,100,222),(98,224,107,221,104,218,101,227),(99,217,108,226,105,223,102,220),(109,240,118,237,115,234,112,231),(110,233,119,230,116,239,113,236),(111,238,120,235,117,232,114,229)]])
150 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 10A | 10B | 10C | 10D | 10E | ··· | 10P | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | ··· | 20T | 30A | 30B | 30C | 30D | 30E | ··· | 30P | 40A | ··· | 40P | 40Q | ··· | 40AN | 60A | ··· | 60H | 60I | ··· | 60T |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | 30 | 30 | 30 | 30 | ··· | 30 | 40 | ··· | 40 | 40 | ··· | 40 | 60 | ··· | 60 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 3 | 3 | 3 | 3 | 6 | ··· | 6 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 3 | ··· | 3 | 6 | ··· | 6 | 2 | ··· | 2 | 4 | ··· | 4 |
150 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | - | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C5 | C10 | C10 | C10 | C20 | C20 | S3 | D6 | Dic3 | Dic3 | C5×S3 | C8○D4 | S3×C10 | C5×Dic3 | C5×Dic3 | C5×C8○D4 | D4.Dic3 | C5×D4.Dic3 |
| kernel | C5×D4.Dic3 | C10×C3⋊C8 | C5×C4.Dic3 | C15×C4○D4 | D4×C15 | Q8×C15 | D4.Dic3 | C2×C3⋊C8 | C4.Dic3 | C3×C4○D4 | C3×D4 | C3×Q8 | C5×C4○D4 | C2×C20 | C5×D4 | C5×Q8 | C4○D4 | C15 | C2×C4 | D4 | Q8 | C3 | C5 | C1 |
| # reps | 1 | 3 | 3 | 1 | 6 | 2 | 4 | 12 | 12 | 4 | 24 | 8 | 1 | 3 | 3 | 1 | 4 | 4 | 12 | 12 | 4 | 16 | 2 | 8 |
Matrix representation of C5×D4.Dic3 ►in GL4(𝔽241) generated by
| 87 | 0 | 0 | 0 |
| 0 | 87 | 0 | 0 |
| 0 | 0 | 91 | 0 |
| 0 | 0 | 0 | 91 |
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 177 | 0 |
| 0 | 0 | 118 | 64 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 177 | 192 |
| 0 | 0 | 118 | 64 |
| 1 | 1 | 0 | 0 |
| 240 | 0 | 0 | 0 |
| 0 | 0 | 177 | 0 |
| 0 | 0 | 0 | 177 |
| 141 | 76 | 0 | 0 |
| 176 | 100 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 0 | 30 |
G:=sub<GL(4,GF(241))| [87,0,0,0,0,87,0,0,0,0,91,0,0,0,0,91],[240,0,0,0,0,240,0,0,0,0,177,118,0,0,0,64],[1,0,0,0,0,1,0,0,0,0,177,118,0,0,192,64],[1,240,0,0,1,0,0,0,0,0,177,0,0,0,0,177],[141,176,0,0,76,100,0,0,0,0,30,0,0,0,0,30] >;
C5×D4.Dic3 in GAP, Magma, Sage, TeX
C_5\times D_4.{\rm Dic}_3 % in TeX
G:=Group("C5xD4.Dic3"); // GroupNames label
G:=SmallGroup(480,827);
// by ID
G=gap.SmallGroup(480,827);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,280,891,102,15686]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=c^2=1,d^6=b^2,e^2=b^2*d^3,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^5>;
// generators/relations