direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C23.21D10, (C2×C60)⋊23C4, (C2×C20)⋊11C12, C4⋊Dic5⋊17C6, (C2×C12)⋊9Dic5, C60.252(C2×C4), C20.59(C2×C12), (C4×Dic5)⋊15C6, C4.15(C6×Dic5), C23.26(C6×D5), C23.D5.5C6, (C12×Dic5)⋊33C2, (C2×C12).453D10, (C22×C60).21C2, (C22×C20).14C6, C12.54(C2×Dic5), (C22×C12).17D5, C15⋊31(C42⋊C2), C30.193(C4○D4), C6.123(C4○D20), C22.6(C6×Dic5), C10.37(C22×C12), (C2×C60).513C22, (C2×C30).361C23, C30.222(C22×C4), (C22×C6).106D10, C6.34(C22×Dic5), (C22×C30).157C22, (C6×Dic5).247C22, C2.5(C2×C6×Dic5), C5⋊5(C3×C42⋊C2), (C2×C4)⋊4(C3×Dic5), C2.4(C3×C4○D20), C22.22(D5×C2×C6), (C2×C20).96(C2×C6), (C3×C4⋊Dic5)⋊35C2, C10.14(C3×C4○D4), (C2×C4).103(C6×D5), (C22×C4).9(C3×D5), (C2×C30).191(C2×C4), (C2×C10).55(C2×C12), (C2×C6).24(C2×Dic5), (C2×C10).44(C22×C6), (C22×C10).44(C2×C6), (C2×Dic5).37(C2×C6), (C2×C6).357(C22×D5), (C3×C23.D5).11C2, SmallGroup(480,719)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C23.21D10
G = < a,b,c,d,e,f | a3=b2=c2=d2=1, e10=d, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >
Subgroups: 336 in 152 conjugacy classes, 98 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, C2×C4, C23, C10, C10, C10, C12, C12, C2×C6, C2×C6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, Dic5, C20, C2×C10, C2×C10, C2×C10, C2×C12, C2×C12, C2×C12, C22×C6, C30, C30, C30, C42⋊C2, C2×Dic5, C2×C20, C2×C20, C22×C10, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C3×Dic5, C60, C2×C30, C2×C30, C2×C30, C4×Dic5, C4⋊Dic5, C23.D5, C22×C20, C3×C42⋊C2, C6×Dic5, C2×C60, C2×C60, C22×C30, C23.21D10, C12×Dic5, C3×C4⋊Dic5, C3×C23.D5, C22×C60, C3×C23.21D10
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, D5, C12, C2×C6, C22×C4, C4○D4, Dic5, D10, C2×C12, C22×C6, C3×D5, C42⋊C2, C2×Dic5, C22×D5, C22×C12, C3×C4○D4, C3×Dic5, C6×D5, C4○D20, C22×Dic5, C3×C42⋊C2, C6×Dic5, D5×C2×C6, C23.21D10, C3×C4○D20, C2×C6×Dic5, C3×C23.21D10
►On 240 points
Generators in S240
(1 188 76)(2 189 77)(3 190 78)(4 191 79)(5 192 80)(6 193 61)(7 194 62)(8 195 63)(9 196 64)(10 197 65)(11 198 66)(12 199 67)(13 200 68)(14 181 69)(15 182 70)(16 183 71)(17 184 72)(18 185 73)(19 186 74)(20 187 75)(21 107 50)(22 108 51)(23 109 52)(24 110 53)(25 111 54)(26 112 55)(27 113 56)(28 114 57)(29 115 58)(30 116 59)(31 117 60)(32 118 41)(33 119 42)(34 120 43)(35 101 44)(36 102 45)(37 103 46)(38 104 47)(39 105 48)(40 106 49)(81 125 235)(82 126 236)(83 127 237)(84 128 238)(85 129 239)(86 130 240)(87 131 221)(88 132 222)(89 133 223)(90 134 224)(91 135 225)(92 136 226)(93 137 227)(94 138 228)(95 139 229)(96 140 230)(97 121 231)(98 122 232)(99 123 233)(100 124 234)(141 166 218)(142 167 219)(143 168 220)(144 169 201)(145 170 202)(146 171 203)(147 172 204)(148 173 205)(149 174 206)(150 175 207)(151 176 208)(152 177 209)(153 178 210)(154 179 211)(155 180 212)(156 161 213)(157 162 214)(158 163 215)(159 164 216)(160 165 217)
(1 115)(2 116)(3 117)(4 118)(5 119)(6 120)(7 101)(8 102)(9 103)(10 104)(11 105)(12 106)(13 107)(14 108)(15 109)(16 110)(17 111)(18 112)(19 113)(20 114)(21 68)(22 69)(23 70)(24 71)(25 72)(26 73)(27 74)(28 75)(29 76)(30 77)(31 78)(32 79)(33 80)(34 61)(35 62)(36 63)(37 64)(38 65)(39 66)(40 67)(41 191)(42 192)(43 193)(44 194)(45 195)(46 196)(47 197)(48 198)(49 199)(50 200)(51 181)(52 182)(53 183)(54 184)(55 185)(56 186)(57 187)(58 188)(59 189)(60 190)(81 174)(82 175)(83 176)(84 177)(85 178)(86 179)(87 180)(88 161)(89 162)(90 163)(91 164)(92 165)(93 166)(94 167)(95 168)(96 169)(97 170)(98 171)(99 172)(100 173)(121 202)(122 203)(123 204)(124 205)(125 206)(126 207)(127 208)(128 209)(129 210)(130 211)(131 212)(132 213)(133 214)(134 215)(135 216)(136 217)(137 218)(138 219)(139 220)(140 201)(141 227)(142 228)(143 229)(144 230)(145 231)(146 232)(147 233)(148 234)(149 235)(150 236)(151 237)(152 238)(153 239)(154 240)(155 221)(156 222)(157 223)(158 224)(159 225)(160 226)
(1 115)(2 116)(3 117)(4 118)(5 119)(6 120)(7 101)(8 102)(9 103)(10 104)(11 105)(12 106)(13 107)(14 108)(15 109)(16 110)(17 111)(18 112)(19 113)(20 114)(21 68)(22 69)(23 70)(24 71)(25 72)(26 73)(27 74)(28 75)(29 76)(30 77)(31 78)(32 79)(33 80)(34 61)(35 62)(36 63)(37 64)(38 65)(39 66)(40 67)(41 191)(42 192)(43 193)(44 194)(45 195)(46 196)(47 197)(48 198)(49 199)(50 200)(51 181)(52 182)(53 183)(54 184)(55 185)(56 186)(57 187)(58 188)(59 189)(60 190)(81 164)(82 165)(83 166)(84 167)(85 168)(86 169)(87 170)(88 171)(89 172)(90 173)(91 174)(92 175)(93 176)(94 177)(95 178)(96 179)(97 180)(98 161)(99 162)(100 163)(121 212)(122 213)(123 214)(124 215)(125 216)(126 217)(127 218)(128 219)(129 220)(130 201)(131 202)(132 203)(133 204)(134 205)(135 206)(136 207)(137 208)(138 209)(139 210)(140 211)(141 237)(142 238)(143 239)(144 240)(145 221)(146 222)(147 223)(148 224)(149 225)(150 226)(151 227)(152 228)(153 229)(154 230)(155 231)(156 232)(157 233)(158 234)(159 235)(160 236)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 97)(88 98)(89 99)(90 100)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)(121 131)(122 132)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)(141 151)(142 152)(143 153)(144 154)(145 155)(146 156)(147 157)(148 158)(149 159)(150 160)(161 171)(162 172)(163 173)(164 174)(165 175)(166 176)(167 177)(168 178)(169 179)(170 180)(181 191)(182 192)(183 193)(184 194)(185 195)(186 196)(187 197)(188 198)(189 199)(190 200)(201 211)(202 212)(203 213)(204 214)(205 215)(206 216)(207 217)(208 218)(209 219)(210 220)(221 231)(222 232)(223 233)(224 234)(225 235)(226 236)(227 237)(228 238)(229 239)(230 240)
(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 94 105 167)(2 83 106 176)(3 92 107 165)(4 81 108 174)(5 90 109 163)(6 99 110 172)(7 88 111 161)(8 97 112 170)(9 86 113 179)(10 95 114 168)(11 84 115 177)(12 93 116 166)(13 82 117 175)(14 91 118 164)(15 100 119 173)(16 89 120 162)(17 98 101 171)(18 87 102 180)(19 96 103 169)(20 85 104 178)(21 160 78 226)(22 149 79 235)(23 158 80 224)(24 147 61 233)(25 156 62 222)(26 145 63 231)(27 154 64 240)(28 143 65 229)(29 152 66 238)(30 141 67 227)(31 150 68 236)(32 159 69 225)(33 148 70 234)(34 157 71 223)(35 146 72 232)(36 155 73 221)(37 144 74 230)(38 153 75 239)(39 142 76 228)(40 151 77 237)(41 216 181 135)(42 205 182 124)(43 214 183 133)(44 203 184 122)(45 212 185 131)(46 201 186 140)(47 210 187 129)(48 219 188 138)(49 208 189 127)(50 217 190 136)(51 206 191 125)(52 215 192 134)(53 204 193 123)(54 213 194 132)(55 202 195 121)(56 211 196 130)(57 220 197 139)(58 209 198 128)(59 218 199 137)(60 207 200 126)
G:=sub<Sym(240)| (1,188,76)(2,189,77)(3,190,78)(4,191,79)(5,192,80)(6,193,61)(7,194,62)(8,195,63)(9,196,64)(10,197,65)(11,198,66)(12,199,67)(13,200,68)(14,181,69)(15,182,70)(16,183,71)(17,184,72)(18,185,73)(19,186,74)(20,187,75)(21,107,50)(22,108,51)(23,109,52)(24,110,53)(25,111,54)(26,112,55)(27,113,56)(28,114,57)(29,115,58)(30,116,59)(31,117,60)(32,118,41)(33,119,42)(34,120,43)(35,101,44)(36,102,45)(37,103,46)(38,104,47)(39,105,48)(40,106,49)(81,125,235)(82,126,236)(83,127,237)(84,128,238)(85,129,239)(86,130,240)(87,131,221)(88,132,222)(89,133,223)(90,134,224)(91,135,225)(92,136,226)(93,137,227)(94,138,228)(95,139,229)(96,140,230)(97,121,231)(98,122,232)(99,123,233)(100,124,234)(141,166,218)(142,167,219)(143,168,220)(144,169,201)(145,170,202)(146,171,203)(147,172,204)(148,173,205)(149,174,206)(150,175,207)(151,176,208)(152,177,209)(153,178,210)(154,179,211)(155,180,212)(156,161,213)(157,162,214)(158,163,215)(159,164,216)(160,165,217), (1,115)(2,116)(3,117)(4,118)(5,119)(6,120)(7,101)(8,102)(9,103)(10,104)(11,105)(12,106)(13,107)(14,108)(15,109)(16,110)(17,111)(18,112)(19,113)(20,114)(21,68)(22,69)(23,70)(24,71)(25,72)(26,73)(27,74)(28,75)(29,76)(30,77)(31,78)(32,79)(33,80)(34,61)(35,62)(36,63)(37,64)(38,65)(39,66)(40,67)(41,191)(42,192)(43,193)(44,194)(45,195)(46,196)(47,197)(48,198)(49,199)(50,200)(51,181)(52,182)(53,183)(54,184)(55,185)(56,186)(57,187)(58,188)(59,189)(60,190)(81,174)(82,175)(83,176)(84,177)(85,178)(86,179)(87,180)(88,161)(89,162)(90,163)(91,164)(92,165)(93,166)(94,167)(95,168)(96,169)(97,170)(98,171)(99,172)(100,173)(121,202)(122,203)(123,204)(124,205)(125,206)(126,207)(127,208)(128,209)(129,210)(130,211)(131,212)(132,213)(133,214)(134,215)(135,216)(136,217)(137,218)(138,219)(139,220)(140,201)(141,227)(142,228)(143,229)(144,230)(145,231)(146,232)(147,233)(148,234)(149,235)(150,236)(151,237)(152,238)(153,239)(154,240)(155,221)(156,222)(157,223)(158,224)(159,225)(160,226), (1,115)(2,116)(3,117)(4,118)(5,119)(6,120)(7,101)(8,102)(9,103)(10,104)(11,105)(12,106)(13,107)(14,108)(15,109)(16,110)(17,111)(18,112)(19,113)(20,114)(21,68)(22,69)(23,70)(24,71)(25,72)(26,73)(27,74)(28,75)(29,76)(30,77)(31,78)(32,79)(33,80)(34,61)(35,62)(36,63)(37,64)(38,65)(39,66)(40,67)(41,191)(42,192)(43,193)(44,194)(45,195)(46,196)(47,197)(48,198)(49,199)(50,200)(51,181)(52,182)(53,183)(54,184)(55,185)(56,186)(57,187)(58,188)(59,189)(60,190)(81,164)(82,165)(83,166)(84,167)(85,168)(86,169)(87,170)(88,171)(89,172)(90,173)(91,174)(92,175)(93,176)(94,177)(95,178)(96,179)(97,180)(98,161)(99,162)(100,163)(121,212)(122,213)(123,214)(124,215)(125,216)(126,217)(127,218)(128,219)(129,220)(130,201)(131,202)(132,203)(133,204)(134,205)(135,206)(136,207)(137,208)(138,209)(139,210)(140,211)(141,237)(142,238)(143,239)(144,240)(145,221)(146,222)(147,223)(148,224)(149,225)(150,226)(151,227)(152,228)(153,229)(154,230)(155,231)(156,232)(157,233)(158,234)(159,235)(160,236), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140)(141,151)(142,152)(143,153)(144,154)(145,155)(146,156)(147,157)(148,158)(149,159)(150,160)(161,171)(162,172)(163,173)(164,174)(165,175)(166,176)(167,177)(168,178)(169,179)(170,180)(181,191)(182,192)(183,193)(184,194)(185,195)(186,196)(187,197)(188,198)(189,199)(190,200)(201,211)(202,212)(203,213)(204,214)(205,215)(206,216)(207,217)(208,218)(209,219)(210,220)(221,231)(222,232)(223,233)(224,234)(225,235)(226,236)(227,237)(228,238)(229,239)(230,240), (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), (1,94,105,167)(2,83,106,176)(3,92,107,165)(4,81,108,174)(5,90,109,163)(6,99,110,172)(7,88,111,161)(8,97,112,170)(9,86,113,179)(10,95,114,168)(11,84,115,177)(12,93,116,166)(13,82,117,175)(14,91,118,164)(15,100,119,173)(16,89,120,162)(17,98,101,171)(18,87,102,180)(19,96,103,169)(20,85,104,178)(21,160,78,226)(22,149,79,235)(23,158,80,224)(24,147,61,233)(25,156,62,222)(26,145,63,231)(27,154,64,240)(28,143,65,229)(29,152,66,238)(30,141,67,227)(31,150,68,236)(32,159,69,225)(33,148,70,234)(34,157,71,223)(35,146,72,232)(36,155,73,221)(37,144,74,230)(38,153,75,239)(39,142,76,228)(40,151,77,237)(41,216,181,135)(42,205,182,124)(43,214,183,133)(44,203,184,122)(45,212,185,131)(46,201,186,140)(47,210,187,129)(48,219,188,138)(49,208,189,127)(50,217,190,136)(51,206,191,125)(52,215,192,134)(53,204,193,123)(54,213,194,132)(55,202,195,121)(56,211,196,130)(57,220,197,139)(58,209,198,128)(59,218,199,137)(60,207,200,126)>;
G:=Group( (1,188,76)(2,189,77)(3,190,78)(4,191,79)(5,192,80)(6,193,61)(7,194,62)(8,195,63)(9,196,64)(10,197,65)(11,198,66)(12,199,67)(13,200,68)(14,181,69)(15,182,70)(16,183,71)(17,184,72)(18,185,73)(19,186,74)(20,187,75)(21,107,50)(22,108,51)(23,109,52)(24,110,53)(25,111,54)(26,112,55)(27,113,56)(28,114,57)(29,115,58)(30,116,59)(31,117,60)(32,118,41)(33,119,42)(34,120,43)(35,101,44)(36,102,45)(37,103,46)(38,104,47)(39,105,48)(40,106,49)(81,125,235)(82,126,236)(83,127,237)(84,128,238)(85,129,239)(86,130,240)(87,131,221)(88,132,222)(89,133,223)(90,134,224)(91,135,225)(92,136,226)(93,137,227)(94,138,228)(95,139,229)(96,140,230)(97,121,231)(98,122,232)(99,123,233)(100,124,234)(141,166,218)(142,167,219)(143,168,220)(144,169,201)(145,170,202)(146,171,203)(147,172,204)(148,173,205)(149,174,206)(150,175,207)(151,176,208)(152,177,209)(153,178,210)(154,179,211)(155,180,212)(156,161,213)(157,162,214)(158,163,215)(159,164,216)(160,165,217), (1,115)(2,116)(3,117)(4,118)(5,119)(6,120)(7,101)(8,102)(9,103)(10,104)(11,105)(12,106)(13,107)(14,108)(15,109)(16,110)(17,111)(18,112)(19,113)(20,114)(21,68)(22,69)(23,70)(24,71)(25,72)(26,73)(27,74)(28,75)(29,76)(30,77)(31,78)(32,79)(33,80)(34,61)(35,62)(36,63)(37,64)(38,65)(39,66)(40,67)(41,191)(42,192)(43,193)(44,194)(45,195)(46,196)(47,197)(48,198)(49,199)(50,200)(51,181)(52,182)(53,183)(54,184)(55,185)(56,186)(57,187)(58,188)(59,189)(60,190)(81,174)(82,175)(83,176)(84,177)(85,178)(86,179)(87,180)(88,161)(89,162)(90,163)(91,164)(92,165)(93,166)(94,167)(95,168)(96,169)(97,170)(98,171)(99,172)(100,173)(121,202)(122,203)(123,204)(124,205)(125,206)(126,207)(127,208)(128,209)(129,210)(130,211)(131,212)(132,213)(133,214)(134,215)(135,216)(136,217)(137,218)(138,219)(139,220)(140,201)(141,227)(142,228)(143,229)(144,230)(145,231)(146,232)(147,233)(148,234)(149,235)(150,236)(151,237)(152,238)(153,239)(154,240)(155,221)(156,222)(157,223)(158,224)(159,225)(160,226), (1,115)(2,116)(3,117)(4,118)(5,119)(6,120)(7,101)(8,102)(9,103)(10,104)(11,105)(12,106)(13,107)(14,108)(15,109)(16,110)(17,111)(18,112)(19,113)(20,114)(21,68)(22,69)(23,70)(24,71)(25,72)(26,73)(27,74)(28,75)(29,76)(30,77)(31,78)(32,79)(33,80)(34,61)(35,62)(36,63)(37,64)(38,65)(39,66)(40,67)(41,191)(42,192)(43,193)(44,194)(45,195)(46,196)(47,197)(48,198)(49,199)(50,200)(51,181)(52,182)(53,183)(54,184)(55,185)(56,186)(57,187)(58,188)(59,189)(60,190)(81,164)(82,165)(83,166)(84,167)(85,168)(86,169)(87,170)(88,171)(89,172)(90,173)(91,174)(92,175)(93,176)(94,177)(95,178)(96,179)(97,180)(98,161)(99,162)(100,163)(121,212)(122,213)(123,214)(124,215)(125,216)(126,217)(127,218)(128,219)(129,220)(130,201)(131,202)(132,203)(133,204)(134,205)(135,206)(136,207)(137,208)(138,209)(139,210)(140,211)(141,237)(142,238)(143,239)(144,240)(145,221)(146,222)(147,223)(148,224)(149,225)(150,226)(151,227)(152,228)(153,229)(154,230)(155,231)(156,232)(157,233)(158,234)(159,235)(160,236), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140)(141,151)(142,152)(143,153)(144,154)(145,155)(146,156)(147,157)(148,158)(149,159)(150,160)(161,171)(162,172)(163,173)(164,174)(165,175)(166,176)(167,177)(168,178)(169,179)(170,180)(181,191)(182,192)(183,193)(184,194)(185,195)(186,196)(187,197)(188,198)(189,199)(190,200)(201,211)(202,212)(203,213)(204,214)(205,215)(206,216)(207,217)(208,218)(209,219)(210,220)(221,231)(222,232)(223,233)(224,234)(225,235)(226,236)(227,237)(228,238)(229,239)(230,240), (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), (1,94,105,167)(2,83,106,176)(3,92,107,165)(4,81,108,174)(5,90,109,163)(6,99,110,172)(7,88,111,161)(8,97,112,170)(9,86,113,179)(10,95,114,168)(11,84,115,177)(12,93,116,166)(13,82,117,175)(14,91,118,164)(15,100,119,173)(16,89,120,162)(17,98,101,171)(18,87,102,180)(19,96,103,169)(20,85,104,178)(21,160,78,226)(22,149,79,235)(23,158,80,224)(24,147,61,233)(25,156,62,222)(26,145,63,231)(27,154,64,240)(28,143,65,229)(29,152,66,238)(30,141,67,227)(31,150,68,236)(32,159,69,225)(33,148,70,234)(34,157,71,223)(35,146,72,232)(36,155,73,221)(37,144,74,230)(38,153,75,239)(39,142,76,228)(40,151,77,237)(41,216,181,135)(42,205,182,124)(43,214,183,133)(44,203,184,122)(45,212,185,131)(46,201,186,140)(47,210,187,129)(48,219,188,138)(49,208,189,127)(50,217,190,136)(51,206,191,125)(52,215,192,134)(53,204,193,123)(54,213,194,132)(55,202,195,121)(56,211,196,130)(57,220,197,139)(58,209,198,128)(59,218,199,137)(60,207,200,126) );
G=PermutationGroup([[(1,188,76),(2,189,77),(3,190,78),(4,191,79),(5,192,80),(6,193,61),(7,194,62),(8,195,63),(9,196,64),(10,197,65),(11,198,66),(12,199,67),(13,200,68),(14,181,69),(15,182,70),(16,183,71),(17,184,72),(18,185,73),(19,186,74),(20,187,75),(21,107,50),(22,108,51),(23,109,52),(24,110,53),(25,111,54),(26,112,55),(27,113,56),(28,114,57),(29,115,58),(30,116,59),(31,117,60),(32,118,41),(33,119,42),(34,120,43),(35,101,44),(36,102,45),(37,103,46),(38,104,47),(39,105,48),(40,106,49),(81,125,235),(82,126,236),(83,127,237),(84,128,238),(85,129,239),(86,130,240),(87,131,221),(88,132,222),(89,133,223),(90,134,224),(91,135,225),(92,136,226),(93,137,227),(94,138,228),(95,139,229),(96,140,230),(97,121,231),(98,122,232),(99,123,233),(100,124,234),(141,166,218),(142,167,219),(143,168,220),(144,169,201),(145,170,202),(146,171,203),(147,172,204),(148,173,205),(149,174,206),(150,175,207),(151,176,208),(152,177,209),(153,178,210),(154,179,211),(155,180,212),(156,161,213),(157,162,214),(158,163,215),(159,164,216),(160,165,217)], [(1,115),(2,116),(3,117),(4,118),(5,119),(6,120),(7,101),(8,102),(9,103),(10,104),(11,105),(12,106),(13,107),(14,108),(15,109),(16,110),(17,111),(18,112),(19,113),(20,114),(21,68),(22,69),(23,70),(24,71),(25,72),(26,73),(27,74),(28,75),(29,76),(30,77),(31,78),(32,79),(33,80),(34,61),(35,62),(36,63),(37,64),(38,65),(39,66),(40,67),(41,191),(42,192),(43,193),(44,194),(45,195),(46,196),(47,197),(48,198),(49,199),(50,200),(51,181),(52,182),(53,183),(54,184),(55,185),(56,186),(57,187),(58,188),(59,189),(60,190),(81,174),(82,175),(83,176),(84,177),(85,178),(86,179),(87,180),(88,161),(89,162),(90,163),(91,164),(92,165),(93,166),(94,167),(95,168),(96,169),(97,170),(98,171),(99,172),(100,173),(121,202),(122,203),(123,204),(124,205),(125,206),(126,207),(127,208),(128,209),(129,210),(130,211),(131,212),(132,213),(133,214),(134,215),(135,216),(136,217),(137,218),(138,219),(139,220),(140,201),(141,227),(142,228),(143,229),(144,230),(145,231),(146,232),(147,233),(148,234),(149,235),(150,236),(151,237),(152,238),(153,239),(154,240),(155,221),(156,222),(157,223),(158,224),(159,225),(160,226)], [(1,115),(2,116),(3,117),(4,118),(5,119),(6,120),(7,101),(8,102),(9,103),(10,104),(11,105),(12,106),(13,107),(14,108),(15,109),(16,110),(17,111),(18,112),(19,113),(20,114),(21,68),(22,69),(23,70),(24,71),(25,72),(26,73),(27,74),(28,75),(29,76),(30,77),(31,78),(32,79),(33,80),(34,61),(35,62),(36,63),(37,64),(38,65),(39,66),(40,67),(41,191),(42,192),(43,193),(44,194),(45,195),(46,196),(47,197),(48,198),(49,199),(50,200),(51,181),(52,182),(53,183),(54,184),(55,185),(56,186),(57,187),(58,188),(59,189),(60,190),(81,164),(82,165),(83,166),(84,167),(85,168),(86,169),(87,170),(88,171),(89,172),(90,173),(91,174),(92,175),(93,176),(94,177),(95,178),(96,179),(97,180),(98,161),(99,162),(100,163),(121,212),(122,213),(123,214),(124,215),(125,216),(126,217),(127,218),(128,219),(129,220),(130,201),(131,202),(132,203),(133,204),(134,205),(135,206),(136,207),(137,208),(138,209),(139,210),(140,211),(141,237),(142,238),(143,239),(144,240),(145,221),(146,222),(147,223),(148,224),(149,225),(150,226),(151,227),(152,228),(153,229),(154,230),(155,231),(156,232),(157,233),(158,234),(159,235),(160,236)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80),(81,91),(82,92),(83,93),(84,94),(85,95),(86,96),(87,97),(88,98),(89,99),(90,100),(101,111),(102,112),(103,113),(104,114),(105,115),(106,116),(107,117),(108,118),(109,119),(110,120),(121,131),(122,132),(123,133),(124,134),(125,135),(126,136),(127,137),(128,138),(129,139),(130,140),(141,151),(142,152),(143,153),(144,154),(145,155),(146,156),(147,157),(148,158),(149,159),(150,160),(161,171),(162,172),(163,173),(164,174),(165,175),(166,176),(167,177),(168,178),(169,179),(170,180),(181,191),(182,192),(183,193),(184,194),(185,195),(186,196),(187,197),(188,198),(189,199),(190,200),(201,211),(202,212),(203,213),(204,214),(205,215),(206,216),(207,217),(208,218),(209,219),(210,220),(221,231),(222,232),(223,233),(224,234),(225,235),(226,236),(227,237),(228,238),(229,239),(230,240)], [(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)], [(1,94,105,167),(2,83,106,176),(3,92,107,165),(4,81,108,174),(5,90,109,163),(6,99,110,172),(7,88,111,161),(8,97,112,170),(9,86,113,179),(10,95,114,168),(11,84,115,177),(12,93,116,166),(13,82,117,175),(14,91,118,164),(15,100,119,173),(16,89,120,162),(17,98,101,171),(18,87,102,180),(19,96,103,169),(20,85,104,178),(21,160,78,226),(22,149,79,235),(23,158,80,224),(24,147,61,233),(25,156,62,222),(26,145,63,231),(27,154,64,240),(28,143,65,229),(29,152,66,238),(30,141,67,227),(31,150,68,236),(32,159,69,225),(33,148,70,234),(34,157,71,223),(35,146,72,232),(36,155,73,221),(37,144,74,230),(38,153,75,239),(39,142,76,228),(40,151,77,237),(41,216,181,135),(42,205,182,124),(43,214,183,133),(44,203,184,122),(45,212,185,131),(46,201,186,140),(47,210,187,129),(48,219,188,138),(49,208,189,127),(50,217,190,136),(51,206,191,125),(52,215,192,134),(53,204,193,123),(54,213,194,132),(55,202,195,121),(56,211,196,130),(57,220,197,139),(58,209,198,128),(59,218,199,137),(60,207,200,126)]])
156 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 5A | 5B | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 10A | ··· | 10N | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 12M | ··· | 12AB | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 30A | ··· | 30AB | 60A | ··· | 60AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 10 | ··· | 10 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
156 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 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C12 | D5 | C4○D4 | Dic5 | D10 | D10 | C3×D5 | C3×C4○D4 | C3×Dic5 | C6×D5 | C6×D5 | C4○D20 | C3×C4○D20 |
| kernel | C3×C23.21D10 | C12×Dic5 | C3×C4⋊Dic5 | C3×C23.D5 | C22×C60 | C23.21D10 | C2×C60 | C4×Dic5 | C4⋊Dic5 | C23.D5 | C22×C20 | C2×C20 | C22×C12 | C30 | C2×C12 | C2×C12 | C22×C6 | C22×C4 | C10 | C2×C4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 2 | 8 | 4 | 4 | 4 | 2 | 16 | 2 | 4 | 8 | 4 | 2 | 4 | 8 | 16 | 8 | 4 | 16 | 32 |
Matrix representation of C3×C23.21D10 ►in GL3(𝔽61) generated by
| 1 | 0 | 0 |
| 0 | 13 | 0 |
| 0 | 0 | 13 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 60 |
| 60 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 60 | 0 |
| 0 | 0 | 60 |
| 60 | 0 | 0 |
| 0 | 28 | 0 |
| 0 | 0 | 37 |
| 11 | 0 | 0 |
| 0 | 0 | 37 |
| 0 | 28 | 0 |
G:=sub<GL(3,GF(61))| [1,0,0,0,13,0,0,0,13],[1,0,0,0,1,0,0,0,60],[60,0,0,0,1,0,0,0,1],[1,0,0,0,60,0,0,0,60],[60,0,0,0,28,0,0,0,37],[11,0,0,0,0,28,0,37,0] >;
C3×C23.21D10 in GAP, Magma, Sage, TeX
C_3\times C_2^3._{21}D_{10} % in TeX
G:=Group("C3xC2^3.21D10"); // GroupNames label
G:=SmallGroup(480,719);
// by ID
G=gap.SmallGroup(480,719);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,168,344,1094,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=1,e^10=d,f^2=d*c=c*d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^9>;
// generators/relations