direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C23.18D10, (C6×D4).11D5, C10.51(C6×D4), (C2×C30).84D4, C23.D5⋊8C6, (D4×C10).10C6, (D4×C30).22C2, C30.405(C2×D4), C23.22(C6×D5), (C22×C6).7D10, C10.D4⋊14C6, (C2×C12).240D10, (C22×Dic5)⋊8C6, C30.238(C4○D4), (C2×C60).423C22, (C2×C30).367C23, C6.120(D4⋊2D5), C15⋊38(C22.D4), (C22×C30).104C22, (C6×Dic5).250C22, (C2×C6×Dic5)⋊16C2, (C2×D4).5(C3×D5), (C2×C4).14(C6×D5), (C2×C10).7(C3×D4), C2.11(C6×C5⋊D4), C22.57(D5×C2×C6), (C2×C20).61(C2×C6), C10.28(C3×C4○D4), C6.132(C2×C5⋊D4), C22.4(C3×C5⋊D4), C5⋊5(C3×C22.D4), C2.15(C3×D4⋊2D5), (C2×C6).40(C5⋊D4), (C3×C23.D5)⋊24C2, (C3×C10.D4)⋊36C2, (C2×C10).50(C22×C6), (C22×C10).23(C2×C6), (C2×Dic5).39(C2×C6), (C2×C6).363(C22×D5), SmallGroup(480,728)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C23.18D10
G = < a,b,c,d,e,f | a3=b2=c2=d2=e10=1, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=fbf-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce-1 >
Subgroups: 416 in 156 conjugacy classes, 66 normal (34 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, D4, C23, C10, C10, C10, C12, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, C2×C10, C2×C10, C2×C10, C2×C12, C2×C12, C3×D4, C22×C6, C30, C30, C30, C22.D4, C2×Dic5, C2×Dic5, C2×C20, C5×D4, C22×C10, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C3×Dic5, C60, C2×C30, C2×C30, C2×C30, C10.D4, C23.D5, C23.D5, C22×Dic5, D4×C10, C3×C22.D4, C6×Dic5, C6×Dic5, C2×C60, D4×C15, C22×C30, C23.18D10, C3×C10.D4, C3×C23.D5, C3×C23.D5, C2×C6×Dic5, D4×C30, C3×C23.18D10
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, C4○D4, D10, C3×D4, C22×C6, C3×D5, C22.D4, C5⋊D4, C22×D5, C6×D4, C3×C4○D4, C6×D5, D4⋊2D5, C2×C5⋊D4, C3×C22.D4, C3×C5⋊D4, D5×C2×C6, C23.18D10, C3×D4⋊2D5, C6×C5⋊D4, C3×C23.18D10
►On 240 points
Generators in S240
(1 33 23)(2 34 24)(3 35 25)(4 36 26)(5 37 27)(6 38 28)(7 39 29)(8 40 30)(9 31 21)(10 32 22)(11 83 73)(12 84 74)(13 85 75)(14 86 76)(15 87 77)(16 88 78)(17 89 79)(18 90 80)(19 81 71)(20 82 72)(41 140 51)(42 131 52)(43 132 53)(44 133 54)(45 134 55)(46 135 56)(47 136 57)(48 137 58)(49 138 59)(50 139 60)(61 164 174)(62 165 175)(63 166 176)(64 167 177)(65 168 178)(66 169 179)(67 170 180)(68 161 171)(69 162 172)(70 163 173)(91 111 101)(92 112 102)(93 113 103)(94 114 104)(95 115 105)(96 116 106)(97 117 107)(98 118 108)(99 119 109)(100 120 110)(121 209 199)(122 210 200)(123 201 191)(124 202 192)(125 203 193)(126 204 194)(127 205 195)(128 206 196)(129 207 197)(130 208 198)(141 151 189)(142 152 190)(143 153 181)(144 154 182)(145 155 183)(146 156 184)(147 157 185)(148 158 186)(149 159 187)(150 160 188)(211 231 221)(212 232 222)(213 233 223)(214 234 224)(215 235 225)(216 236 226)(217 237 227)(218 238 228)(219 239 229)(220 240 230)
(2 98)(4 100)(6 92)(8 94)(10 96)(12 133)(14 135)(16 137)(18 139)(20 131)(22 106)(24 108)(26 110)(28 102)(30 104)(32 116)(34 118)(36 120)(38 112)(40 114)(42 72)(44 74)(46 76)(48 78)(50 80)(52 82)(54 84)(56 86)(58 88)(60 90)(62 185)(64 187)(66 189)(68 181)(70 183)(122 218)(124 220)(126 212)(128 214)(130 216)(141 169)(143 161)(145 163)(147 165)(149 167)(151 179)(153 171)(155 173)(157 175)(159 177)(192 230)(194 222)(196 224)(198 226)(200 228)(202 240)(204 232)(206 234)(208 236)(210 238)
(1 75)(2 76)(3 77)(4 78)(5 79)(6 80)(7 71)(8 72)(9 73)(10 74)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(21 83)(22 84)(23 85)(24 86)(25 87)(26 88)(27 89)(28 90)(29 81)(30 82)(41 93)(42 94)(43 95)(44 96)(45 97)(46 98)(47 99)(48 100)(49 91)(50 92)(51 103)(52 104)(53 105)(54 106)(55 107)(56 108)(57 109)(58 110)(59 101)(60 102)(61 237)(62 238)(63 239)(64 240)(65 231)(66 232)(67 233)(68 234)(69 235)(70 236)(111 138)(112 139)(113 140)(114 131)(115 132)(116 133)(117 134)(118 135)(119 136)(120 137)(121 156)(122 157)(123 158)(124 159)(125 160)(126 151)(127 152)(128 153)(129 154)(130 155)(141 194)(142 195)(143 196)(144 197)(145 198)(146 199)(147 200)(148 191)(149 192)(150 193)(161 224)(162 225)(163 226)(164 227)(165 228)(166 229)(167 230)(168 221)(169 222)(170 223)(171 214)(172 215)(173 216)(174 217)(175 218)(176 219)(177 220)(178 211)(179 212)(180 213)(181 206)(182 207)(183 208)(184 209)(185 210)(186 201)(187 202)(188 203)(189 204)(190 205)
(1 97)(2 98)(3 99)(4 100)(5 91)(6 92)(7 93)(8 94)(9 95)(10 96)(11 132)(12 133)(13 134)(14 135)(15 136)(16 137)(17 138)(18 139)(19 140)(20 131)(21 105)(22 106)(23 107)(24 108)(25 109)(26 110)(27 101)(28 102)(29 103)(30 104)(31 115)(32 116)(33 117)(34 118)(35 119)(36 120)(37 111)(38 112)(39 113)(40 114)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(51 81)(52 82)(53 83)(54 84)(55 85)(56 86)(57 87)(58 88)(59 89)(60 90)(61 184)(62 185)(63 186)(64 187)(65 188)(66 189)(67 190)(68 181)(69 182)(70 183)(121 217)(122 218)(123 219)(124 220)(125 211)(126 212)(127 213)(128 214)(129 215)(130 216)(141 169)(142 170)(143 161)(144 162)(145 163)(146 164)(147 165)(148 166)(149 167)(150 168)(151 179)(152 180)(153 171)(154 172)(155 173)(156 174)(157 175)(158 176)(159 177)(160 178)(191 229)(192 230)(193 221)(194 222)(195 223)(196 224)(197 225)(198 226)(199 227)(200 228)(201 239)(202 240)(203 231)(204 232)(205 233)(206 234)(207 235)(208 236)(209 237)(210 238)
(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)
(1 179 45 126)(2 211 46 160)(3 177 47 124)(4 219 48 158)(5 175 49 122)(6 217 50 156)(7 173 41 130)(8 215 42 154)(9 171 43 128)(10 213 44 152)(11 234 115 181)(12 67 116 205)(13 232 117 189)(14 65 118 203)(15 240 119 187)(16 63 120 201)(17 238 111 185)(18 61 112 209)(19 236 113 183)(20 69 114 207)(21 161 53 196)(22 223 54 142)(23 169 55 194)(24 221 56 150)(25 167 57 192)(26 229 58 148)(27 165 59 200)(28 227 60 146)(29 163 51 198)(30 225 52 144)(31 68 132 206)(32 233 133 190)(33 66 134 204)(34 231 135 188)(35 64 136 202)(36 239 137 186)(37 62 138 210)(38 237 139 184)(39 70 140 208)(40 235 131 182)(71 216 93 155)(72 172 94 129)(73 214 95 153)(74 180 96 127)(75 212 97 151)(76 178 98 125)(77 220 99 159)(78 176 100 123)(79 218 91 157)(80 174 92 121)(81 226 103 145)(82 162 104 197)(83 224 105 143)(84 170 106 195)(85 222 107 141)(86 168 108 193)(87 230 109 149)(88 166 110 191)(89 228 101 147)(90 164 102 199)
G:=sub<Sym(240)| (1,33,23)(2,34,24)(3,35,25)(4,36,26)(5,37,27)(6,38,28)(7,39,29)(8,40,30)(9,31,21)(10,32,22)(11,83,73)(12,84,74)(13,85,75)(14,86,76)(15,87,77)(16,88,78)(17,89,79)(18,90,80)(19,81,71)(20,82,72)(41,140,51)(42,131,52)(43,132,53)(44,133,54)(45,134,55)(46,135,56)(47,136,57)(48,137,58)(49,138,59)(50,139,60)(61,164,174)(62,165,175)(63,166,176)(64,167,177)(65,168,178)(66,169,179)(67,170,180)(68,161,171)(69,162,172)(70,163,173)(91,111,101)(92,112,102)(93,113,103)(94,114,104)(95,115,105)(96,116,106)(97,117,107)(98,118,108)(99,119,109)(100,120,110)(121,209,199)(122,210,200)(123,201,191)(124,202,192)(125,203,193)(126,204,194)(127,205,195)(128,206,196)(129,207,197)(130,208,198)(141,151,189)(142,152,190)(143,153,181)(144,154,182)(145,155,183)(146,156,184)(147,157,185)(148,158,186)(149,159,187)(150,160,188)(211,231,221)(212,232,222)(213,233,223)(214,234,224)(215,235,225)(216,236,226)(217,237,227)(218,238,228)(219,239,229)(220,240,230), (2,98)(4,100)(6,92)(8,94)(10,96)(12,133)(14,135)(16,137)(18,139)(20,131)(22,106)(24,108)(26,110)(28,102)(30,104)(32,116)(34,118)(36,120)(38,112)(40,114)(42,72)(44,74)(46,76)(48,78)(50,80)(52,82)(54,84)(56,86)(58,88)(60,90)(62,185)(64,187)(66,189)(68,181)(70,183)(122,218)(124,220)(126,212)(128,214)(130,216)(141,169)(143,161)(145,163)(147,165)(149,167)(151,179)(153,171)(155,173)(157,175)(159,177)(192,230)(194,222)(196,224)(198,226)(200,228)(202,240)(204,232)(206,234)(208,236)(210,238), (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,71)(8,72)(9,73)(10,74)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,83)(22,84)(23,85)(24,86)(25,87)(26,88)(27,89)(28,90)(29,81)(30,82)(41,93)(42,94)(43,95)(44,96)(45,97)(46,98)(47,99)(48,100)(49,91)(50,92)(51,103)(52,104)(53,105)(54,106)(55,107)(56,108)(57,109)(58,110)(59,101)(60,102)(61,237)(62,238)(63,239)(64,240)(65,231)(66,232)(67,233)(68,234)(69,235)(70,236)(111,138)(112,139)(113,140)(114,131)(115,132)(116,133)(117,134)(118,135)(119,136)(120,137)(121,156)(122,157)(123,158)(124,159)(125,160)(126,151)(127,152)(128,153)(129,154)(130,155)(141,194)(142,195)(143,196)(144,197)(145,198)(146,199)(147,200)(148,191)(149,192)(150,193)(161,224)(162,225)(163,226)(164,227)(165,228)(166,229)(167,230)(168,221)(169,222)(170,223)(171,214)(172,215)(173,216)(174,217)(175,218)(176,219)(177,220)(178,211)(179,212)(180,213)(181,206)(182,207)(183,208)(184,209)(185,210)(186,201)(187,202)(188,203)(189,204)(190,205), (1,97)(2,98)(3,99)(4,100)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,132)(12,133)(13,134)(14,135)(15,136)(16,137)(17,138)(18,139)(19,140)(20,131)(21,105)(22,106)(23,107)(24,108)(25,109)(26,110)(27,101)(28,102)(29,103)(30,104)(31,115)(32,116)(33,117)(34,118)(35,119)(36,120)(37,111)(38,112)(39,113)(40,114)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,81)(52,82)(53,83)(54,84)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90)(61,184)(62,185)(63,186)(64,187)(65,188)(66,189)(67,190)(68,181)(69,182)(70,183)(121,217)(122,218)(123,219)(124,220)(125,211)(126,212)(127,213)(128,214)(129,215)(130,216)(141,169)(142,170)(143,161)(144,162)(145,163)(146,164)(147,165)(148,166)(149,167)(150,168)(151,179)(152,180)(153,171)(154,172)(155,173)(156,174)(157,175)(158,176)(159,177)(160,178)(191,229)(192,230)(193,221)(194,222)(195,223)(196,224)(197,225)(198,226)(199,227)(200,228)(201,239)(202,240)(203,231)(204,232)(205,233)(206,234)(207,235)(208,236)(209,237)(210,238), (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), (1,179,45,126)(2,211,46,160)(3,177,47,124)(4,219,48,158)(5,175,49,122)(6,217,50,156)(7,173,41,130)(8,215,42,154)(9,171,43,128)(10,213,44,152)(11,234,115,181)(12,67,116,205)(13,232,117,189)(14,65,118,203)(15,240,119,187)(16,63,120,201)(17,238,111,185)(18,61,112,209)(19,236,113,183)(20,69,114,207)(21,161,53,196)(22,223,54,142)(23,169,55,194)(24,221,56,150)(25,167,57,192)(26,229,58,148)(27,165,59,200)(28,227,60,146)(29,163,51,198)(30,225,52,144)(31,68,132,206)(32,233,133,190)(33,66,134,204)(34,231,135,188)(35,64,136,202)(36,239,137,186)(37,62,138,210)(38,237,139,184)(39,70,140,208)(40,235,131,182)(71,216,93,155)(72,172,94,129)(73,214,95,153)(74,180,96,127)(75,212,97,151)(76,178,98,125)(77,220,99,159)(78,176,100,123)(79,218,91,157)(80,174,92,121)(81,226,103,145)(82,162,104,197)(83,224,105,143)(84,170,106,195)(85,222,107,141)(86,168,108,193)(87,230,109,149)(88,166,110,191)(89,228,101,147)(90,164,102,199)>;
G:=Group( (1,33,23)(2,34,24)(3,35,25)(4,36,26)(5,37,27)(6,38,28)(7,39,29)(8,40,30)(9,31,21)(10,32,22)(11,83,73)(12,84,74)(13,85,75)(14,86,76)(15,87,77)(16,88,78)(17,89,79)(18,90,80)(19,81,71)(20,82,72)(41,140,51)(42,131,52)(43,132,53)(44,133,54)(45,134,55)(46,135,56)(47,136,57)(48,137,58)(49,138,59)(50,139,60)(61,164,174)(62,165,175)(63,166,176)(64,167,177)(65,168,178)(66,169,179)(67,170,180)(68,161,171)(69,162,172)(70,163,173)(91,111,101)(92,112,102)(93,113,103)(94,114,104)(95,115,105)(96,116,106)(97,117,107)(98,118,108)(99,119,109)(100,120,110)(121,209,199)(122,210,200)(123,201,191)(124,202,192)(125,203,193)(126,204,194)(127,205,195)(128,206,196)(129,207,197)(130,208,198)(141,151,189)(142,152,190)(143,153,181)(144,154,182)(145,155,183)(146,156,184)(147,157,185)(148,158,186)(149,159,187)(150,160,188)(211,231,221)(212,232,222)(213,233,223)(214,234,224)(215,235,225)(216,236,226)(217,237,227)(218,238,228)(219,239,229)(220,240,230), (2,98)(4,100)(6,92)(8,94)(10,96)(12,133)(14,135)(16,137)(18,139)(20,131)(22,106)(24,108)(26,110)(28,102)(30,104)(32,116)(34,118)(36,120)(38,112)(40,114)(42,72)(44,74)(46,76)(48,78)(50,80)(52,82)(54,84)(56,86)(58,88)(60,90)(62,185)(64,187)(66,189)(68,181)(70,183)(122,218)(124,220)(126,212)(128,214)(130,216)(141,169)(143,161)(145,163)(147,165)(149,167)(151,179)(153,171)(155,173)(157,175)(159,177)(192,230)(194,222)(196,224)(198,226)(200,228)(202,240)(204,232)(206,234)(208,236)(210,238), (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,71)(8,72)(9,73)(10,74)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,83)(22,84)(23,85)(24,86)(25,87)(26,88)(27,89)(28,90)(29,81)(30,82)(41,93)(42,94)(43,95)(44,96)(45,97)(46,98)(47,99)(48,100)(49,91)(50,92)(51,103)(52,104)(53,105)(54,106)(55,107)(56,108)(57,109)(58,110)(59,101)(60,102)(61,237)(62,238)(63,239)(64,240)(65,231)(66,232)(67,233)(68,234)(69,235)(70,236)(111,138)(112,139)(113,140)(114,131)(115,132)(116,133)(117,134)(118,135)(119,136)(120,137)(121,156)(122,157)(123,158)(124,159)(125,160)(126,151)(127,152)(128,153)(129,154)(130,155)(141,194)(142,195)(143,196)(144,197)(145,198)(146,199)(147,200)(148,191)(149,192)(150,193)(161,224)(162,225)(163,226)(164,227)(165,228)(166,229)(167,230)(168,221)(169,222)(170,223)(171,214)(172,215)(173,216)(174,217)(175,218)(176,219)(177,220)(178,211)(179,212)(180,213)(181,206)(182,207)(183,208)(184,209)(185,210)(186,201)(187,202)(188,203)(189,204)(190,205), (1,97)(2,98)(3,99)(4,100)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,132)(12,133)(13,134)(14,135)(15,136)(16,137)(17,138)(18,139)(19,140)(20,131)(21,105)(22,106)(23,107)(24,108)(25,109)(26,110)(27,101)(28,102)(29,103)(30,104)(31,115)(32,116)(33,117)(34,118)(35,119)(36,120)(37,111)(38,112)(39,113)(40,114)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,81)(52,82)(53,83)(54,84)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90)(61,184)(62,185)(63,186)(64,187)(65,188)(66,189)(67,190)(68,181)(69,182)(70,183)(121,217)(122,218)(123,219)(124,220)(125,211)(126,212)(127,213)(128,214)(129,215)(130,216)(141,169)(142,170)(143,161)(144,162)(145,163)(146,164)(147,165)(148,166)(149,167)(150,168)(151,179)(152,180)(153,171)(154,172)(155,173)(156,174)(157,175)(158,176)(159,177)(160,178)(191,229)(192,230)(193,221)(194,222)(195,223)(196,224)(197,225)(198,226)(199,227)(200,228)(201,239)(202,240)(203,231)(204,232)(205,233)(206,234)(207,235)(208,236)(209,237)(210,238), (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), (1,179,45,126)(2,211,46,160)(3,177,47,124)(4,219,48,158)(5,175,49,122)(6,217,50,156)(7,173,41,130)(8,215,42,154)(9,171,43,128)(10,213,44,152)(11,234,115,181)(12,67,116,205)(13,232,117,189)(14,65,118,203)(15,240,119,187)(16,63,120,201)(17,238,111,185)(18,61,112,209)(19,236,113,183)(20,69,114,207)(21,161,53,196)(22,223,54,142)(23,169,55,194)(24,221,56,150)(25,167,57,192)(26,229,58,148)(27,165,59,200)(28,227,60,146)(29,163,51,198)(30,225,52,144)(31,68,132,206)(32,233,133,190)(33,66,134,204)(34,231,135,188)(35,64,136,202)(36,239,137,186)(37,62,138,210)(38,237,139,184)(39,70,140,208)(40,235,131,182)(71,216,93,155)(72,172,94,129)(73,214,95,153)(74,180,96,127)(75,212,97,151)(76,178,98,125)(77,220,99,159)(78,176,100,123)(79,218,91,157)(80,174,92,121)(81,226,103,145)(82,162,104,197)(83,224,105,143)(84,170,106,195)(85,222,107,141)(86,168,108,193)(87,230,109,149)(88,166,110,191)(89,228,101,147)(90,164,102,199) );
G=PermutationGroup([[(1,33,23),(2,34,24),(3,35,25),(4,36,26),(5,37,27),(6,38,28),(7,39,29),(8,40,30),(9,31,21),(10,32,22),(11,83,73),(12,84,74),(13,85,75),(14,86,76),(15,87,77),(16,88,78),(17,89,79),(18,90,80),(19,81,71),(20,82,72),(41,140,51),(42,131,52),(43,132,53),(44,133,54),(45,134,55),(46,135,56),(47,136,57),(48,137,58),(49,138,59),(50,139,60),(61,164,174),(62,165,175),(63,166,176),(64,167,177),(65,168,178),(66,169,179),(67,170,180),(68,161,171),(69,162,172),(70,163,173),(91,111,101),(92,112,102),(93,113,103),(94,114,104),(95,115,105),(96,116,106),(97,117,107),(98,118,108),(99,119,109),(100,120,110),(121,209,199),(122,210,200),(123,201,191),(124,202,192),(125,203,193),(126,204,194),(127,205,195),(128,206,196),(129,207,197),(130,208,198),(141,151,189),(142,152,190),(143,153,181),(144,154,182),(145,155,183),(146,156,184),(147,157,185),(148,158,186),(149,159,187),(150,160,188),(211,231,221),(212,232,222),(213,233,223),(214,234,224),(215,235,225),(216,236,226),(217,237,227),(218,238,228),(219,239,229),(220,240,230)], [(2,98),(4,100),(6,92),(8,94),(10,96),(12,133),(14,135),(16,137),(18,139),(20,131),(22,106),(24,108),(26,110),(28,102),(30,104),(32,116),(34,118),(36,120),(38,112),(40,114),(42,72),(44,74),(46,76),(48,78),(50,80),(52,82),(54,84),(56,86),(58,88),(60,90),(62,185),(64,187),(66,189),(68,181),(70,183),(122,218),(124,220),(126,212),(128,214),(130,216),(141,169),(143,161),(145,163),(147,165),(149,167),(151,179),(153,171),(155,173),(157,175),(159,177),(192,230),(194,222),(196,224),(198,226),(200,228),(202,240),(204,232),(206,234),(208,236),(210,238)], [(1,75),(2,76),(3,77),(4,78),(5,79),(6,80),(7,71),(8,72),(9,73),(10,74),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(21,83),(22,84),(23,85),(24,86),(25,87),(26,88),(27,89),(28,90),(29,81),(30,82),(41,93),(42,94),(43,95),(44,96),(45,97),(46,98),(47,99),(48,100),(49,91),(50,92),(51,103),(52,104),(53,105),(54,106),(55,107),(56,108),(57,109),(58,110),(59,101),(60,102),(61,237),(62,238),(63,239),(64,240),(65,231),(66,232),(67,233),(68,234),(69,235),(70,236),(111,138),(112,139),(113,140),(114,131),(115,132),(116,133),(117,134),(118,135),(119,136),(120,137),(121,156),(122,157),(123,158),(124,159),(125,160),(126,151),(127,152),(128,153),(129,154),(130,155),(141,194),(142,195),(143,196),(144,197),(145,198),(146,199),(147,200),(148,191),(149,192),(150,193),(161,224),(162,225),(163,226),(164,227),(165,228),(166,229),(167,230),(168,221),(169,222),(170,223),(171,214),(172,215),(173,216),(174,217),(175,218),(176,219),(177,220),(178,211),(179,212),(180,213),(181,206),(182,207),(183,208),(184,209),(185,210),(186,201),(187,202),(188,203),(189,204),(190,205)], [(1,97),(2,98),(3,99),(4,100),(5,91),(6,92),(7,93),(8,94),(9,95),(10,96),(11,132),(12,133),(13,134),(14,135),(15,136),(16,137),(17,138),(18,139),(19,140),(20,131),(21,105),(22,106),(23,107),(24,108),(25,109),(26,110),(27,101),(28,102),(29,103),(30,104),(31,115),(32,116),(33,117),(34,118),(35,119),(36,120),(37,111),(38,112),(39,113),(40,114),(41,71),(42,72),(43,73),(44,74),(45,75),(46,76),(47,77),(48,78),(49,79),(50,80),(51,81),(52,82),(53,83),(54,84),(55,85),(56,86),(57,87),(58,88),(59,89),(60,90),(61,184),(62,185),(63,186),(64,187),(65,188),(66,189),(67,190),(68,181),(69,182),(70,183),(121,217),(122,218),(123,219),(124,220),(125,211),(126,212),(127,213),(128,214),(129,215),(130,216),(141,169),(142,170),(143,161),(144,162),(145,163),(146,164),(147,165),(148,166),(149,167),(150,168),(151,179),(152,180),(153,171),(154,172),(155,173),(156,174),(157,175),(158,176),(159,177),(160,178),(191,229),(192,230),(193,221),(194,222),(195,223),(196,224),(197,225),(198,226),(199,227),(200,228),(201,239),(202,240),(203,231),(204,232),(205,233),(206,234),(207,235),(208,236),(209,237),(210,238)], [(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)], [(1,179,45,126),(2,211,46,160),(3,177,47,124),(4,219,48,158),(5,175,49,122),(6,217,50,156),(7,173,41,130),(8,215,42,154),(9,171,43,128),(10,213,44,152),(11,234,115,181),(12,67,116,205),(13,232,117,189),(14,65,118,203),(15,240,119,187),(16,63,120,201),(17,238,111,185),(18,61,112,209),(19,236,113,183),(20,69,114,207),(21,161,53,196),(22,223,54,142),(23,169,55,194),(24,221,56,150),(25,167,57,192),(26,229,58,148),(27,165,59,200),(28,227,60,146),(29,163,51,198),(30,225,52,144),(31,68,132,206),(32,233,133,190),(33,66,134,204),(34,231,135,188),(35,64,136,202),(36,239,137,186),(37,62,138,210),(38,237,139,184),(39,70,140,208),(40,235,131,182),(71,216,93,155),(72,172,94,129),(73,214,95,153),(74,180,96,127),(75,212,97,151),(76,178,98,125),(77,220,99,159),(78,176,100,123),(79,218,91,157),(80,174,92,121),(81,226,103,145),(82,162,104,197),(83,224,105,143),(84,170,106,195),(85,222,107,141),(86,168,108,193),(87,230,109,149),(88,166,110,191),(89,228,101,147),(90,164,102,199)]])
102 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | ··· | 12J | 12K | 12L | 12M | 12N | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 30A | ··· | 30L | 30M | ··· | 30AB | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 1 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
102 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | D4 | D5 | C4○D4 | D10 | D10 | C3×D4 | C3×D5 | C5⋊D4 | C3×C4○D4 | C6×D5 | C6×D5 | C3×C5⋊D4 | D4⋊2D5 | C3×D4⋊2D5 |
| kernel | C3×C23.18D10 | C3×C10.D4 | C3×C23.D5 | C2×C6×Dic5 | D4×C30 | C23.18D10 | C10.D4 | C23.D5 | C22×Dic5 | D4×C10 | C2×C30 | C6×D4 | C30 | C2×C12 | C22×C6 | C2×C10 | C2×D4 | C2×C6 | C10 | C2×C4 | C23 | C22 | C6 | C2 |
| # reps | 1 | 2 | 3 | 1 | 1 | 2 | 4 | 6 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 4 | 8 | 8 | 4 | 8 | 16 | 4 | 8 |
Matrix representation of C3×C23.18D10 ►in GL4(𝔽61) generated by
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 60 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 53 | 22 | 0 | 0 |
| 9 | 44 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 34 | 58 | 0 | 0 |
| 40 | 27 | 0 | 0 |
| 0 | 0 | 0 | 50 |
| 0 | 0 | 11 | 0 |
G:=sub<GL(4,GF(61))| [13,0,0,0,0,13,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[53,9,0,0,22,44,0,0,0,0,0,1,0,0,1,0],[34,40,0,0,58,27,0,0,0,0,0,11,0,0,50,0] >;
C3×C23.18D10 in GAP, Magma, Sage, TeX
C_3\times C_2^3._{18}D_{10} % in TeX
G:=Group("C3xC2^3.18D10"); // GroupNames label
G:=SmallGroup(480,728);
// by ID
G=gap.SmallGroup(480,728);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,336,590,555,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=e^10=1,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,e*b*e^-1=f*b*f^-1=b*d=d*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*e^-1>;
// generators/relations