direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C23.23D10, (C22×C20)⋊6C6, (C22×C60)⋊2C2, C10.46(C6×D4), (C22×C12)⋊3D5, C23.D5⋊6C6, D10⋊C4⋊2C6, C10.D4⋊3C6, C30.400(C2×D4), (C2×C30).162D4, C23.28(C6×D5), (C2×C12).378D10, C30.195(C4○D4), C6.125(C4○D20), (C2×C60).448C22, (C2×C30).364C23, (C22×C6).108D10, C15⋊35(C22.D4), (C22×C30).160C22, (C6×Dic5).162C22, C2.6(C6×C5⋊D4), (C2×C4).66(C6×D5), (C2×C5⋊D4).6C6, (C22×C4)⋊5(C3×D5), C22.55(D5×C2×C6), (C2×C20).80(C2×C6), C10.16(C3×C4○D4), C2.18(C3×C4○D20), (C2×C10).37(C3×D4), (C6×C5⋊D4).13C2, C6.127(C2×C5⋊D4), (C3×D10⋊C4)⋊2C2, (D5×C2×C6).81C22, C5⋊4(C3×C22.D4), C22.9(C3×C5⋊D4), (C3×C10.D4)⋊3C2, (C2×C6).62(C5⋊D4), (C3×C23.D5)⋊22C2, (C22×C10).47(C2×C6), (C2×C10).47(C22×C6), (C2×Dic5).14(C2×C6), (C22×D5).11(C2×C6), (C2×C6).360(C22×D5), SmallGroup(480,722)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C23.23D10
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=ce9 >
Subgroups: 480 in 156 conjugacy classes, 66 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C10, C12, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C3×D5, C30, C30, C30, C22.D4, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C3×Dic5, C60, C6×D5, C2×C30, C2×C30, C2×C30, C10.D4, D10⋊C4, C23.D5, C2×C5⋊D4, C22×C20, C3×C22.D4, C6×Dic5, C6×Dic5, C3×C5⋊D4, C2×C60, C2×C60, D5×C2×C6, C22×C30, C23.23D10, C3×C10.D4, C3×D10⋊C4, C3×C23.D5, C6×C5⋊D4, C22×C60, C3×C23.23D10
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, C4○D20, C2×C5⋊D4, C3×C22.D4, C3×C5⋊D4, D5×C2×C6, C23.23D10, C3×C4○D20, C6×C5⋊D4, C3×C23.23D10
►On 240 points
Generators in S240
(1 29 182)(2 30 183)(3 31 184)(4 32 185)(5 33 186)(6 34 187)(7 35 188)(8 36 189)(9 37 190)(10 38 191)(11 39 192)(12 40 193)(13 21 194)(14 22 195)(15 23 196)(16 24 197)(17 25 198)(18 26 199)(19 27 200)(20 28 181)(41 142 135)(42 143 136)(43 144 137)(44 145 138)(45 146 139)(46 147 140)(47 148 121)(48 149 122)(49 150 123)(50 151 124)(51 152 125)(52 153 126)(53 154 127)(54 155 128)(55 156 129)(56 157 130)(57 158 131)(58 159 132)(59 160 133)(60 141 134)(61 220 180)(62 201 161)(63 202 162)(64 203 163)(65 204 164)(66 205 165)(67 206 166)(68 207 167)(69 208 168)(70 209 169)(71 210 170)(72 211 171)(73 212 172)(74 213 173)(75 214 174)(76 215 175)(77 216 176)(78 217 177)(79 218 178)(80 219 179)(81 232 102)(82 233 103)(83 234 104)(84 235 105)(85 236 106)(86 237 107)(87 238 108)(88 239 109)(89 240 110)(90 221 111)(91 222 112)(92 223 113)(93 224 114)(94 225 115)(95 226 116)(96 227 117)(97 228 118)(98 229 119)(99 230 120)(100 231 101)
(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(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)(221 231)(222 232)(223 233)(224 234)(225 235)(226 236)(227 237)(228 238)(229 239)(230 240)
(1 72)(2 73)(3 74)(4 75)(5 76)(6 77)(7 78)(8 79)(9 80)(10 61)(11 62)(12 63)(13 64)(14 65)(15 66)(16 67)(17 68)(18 69)(19 70)(20 71)(21 203)(22 204)(23 205)(24 206)(25 207)(26 208)(27 209)(28 210)(29 211)(30 212)(31 213)(32 214)(33 215)(34 216)(35 217)(36 218)(37 219)(38 220)(39 201)(40 202)(41 115)(42 116)(43 117)(44 118)(45 119)(46 120)(47 101)(48 102)(49 103)(50 104)(51 105)(52 106)(53 107)(54 108)(55 109)(56 110)(57 111)(58 112)(59 113)(60 114)(81 149)(82 150)(83 151)(84 152)(85 153)(86 154)(87 155)(88 156)(89 157)(90 158)(91 159)(92 160)(93 141)(94 142)(95 143)(96 144)(97 145)(98 146)(99 147)(100 148)(121 231)(122 232)(123 233)(124 234)(125 235)(126 236)(127 237)(128 238)(129 239)(130 240)(131 221)(132 222)(133 223)(134 224)(135 225)(136 226)(137 227)(138 228)(139 229)(140 230)(161 192)(162 193)(163 194)(164 195)(165 196)(166 197)(167 198)(168 199)(169 200)(170 181)(171 182)(172 183)(173 184)(174 185)(175 186)(176 187)(177 188)(178 189)(179 190)(180 191)
(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)
(1 225 62 125)(2 124 63 224)(3 223 64 123)(4 122 65 222)(5 221 66 121)(6 140 67 240)(7 239 68 139)(8 138 69 238)(9 237 70 137)(10 136 71 236)(11 235 72 135)(12 134 73 234)(13 233 74 133)(14 132 75 232)(15 231 76 131)(16 130 77 230)(17 229 78 129)(18 128 79 228)(19 227 80 127)(20 126 61 226)(21 103 213 59)(22 58 214 102)(23 101 215 57)(24 56 216 120)(25 119 217 55)(26 54 218 118)(27 117 219 53)(28 52 220 116)(29 115 201 51)(30 50 202 114)(31 113 203 49)(32 48 204 112)(33 111 205 47)(34 46 206 110)(35 109 207 45)(36 44 208 108)(37 107 209 43)(38 42 210 106)(39 105 211 41)(40 60 212 104)(81 195 159 174)(82 173 160 194)(83 193 141 172)(84 171 142 192)(85 191 143 170)(86 169 144 190)(87 189 145 168)(88 167 146 188)(89 187 147 166)(90 165 148 186)(91 185 149 164)(92 163 150 184)(93 183 151 162)(94 161 152 182)(95 181 153 180)(96 179 154 200)(97 199 155 178)(98 177 156 198)(99 197 157 176)(100 175 158 196)
G:=sub<Sym(240)| (1,29,182)(2,30,183)(3,31,184)(4,32,185)(5,33,186)(6,34,187)(7,35,188)(8,36,189)(9,37,190)(10,38,191)(11,39,192)(12,40,193)(13,21,194)(14,22,195)(15,23,196)(16,24,197)(17,25,198)(18,26,199)(19,27,200)(20,28,181)(41,142,135)(42,143,136)(43,144,137)(44,145,138)(45,146,139)(46,147,140)(47,148,121)(48,149,122)(49,150,123)(50,151,124)(51,152,125)(52,153,126)(53,154,127)(54,155,128)(55,156,129)(56,157,130)(57,158,131)(58,159,132)(59,160,133)(60,141,134)(61,220,180)(62,201,161)(63,202,162)(64,203,163)(65,204,164)(66,205,165)(67,206,166)(68,207,167)(69,208,168)(70,209,169)(71,210,170)(72,211,171)(73,212,172)(74,213,173)(75,214,174)(76,215,175)(77,216,176)(78,217,177)(79,218,178)(80,219,179)(81,232,102)(82,233,103)(83,234,104)(84,235,105)(85,236,106)(86,237,107)(87,238,108)(88,239,109)(89,240,110)(90,221,111)(91,222,112)(92,223,113)(93,224,114)(94,225,115)(95,226,116)(96,227,117)(97,228,118)(98,229,119)(99,230,120)(100,231,101), (41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(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)(221,231)(222,232)(223,233)(224,234)(225,235)(226,236)(227,237)(228,238)(229,239)(230,240), (1,72)(2,73)(3,74)(4,75)(5,76)(6,77)(7,78)(8,79)(9,80)(10,61)(11,62)(12,63)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,70)(20,71)(21,203)(22,204)(23,205)(24,206)(25,207)(26,208)(27,209)(28,210)(29,211)(30,212)(31,213)(32,214)(33,215)(34,216)(35,217)(36,218)(37,219)(38,220)(39,201)(40,202)(41,115)(42,116)(43,117)(44,118)(45,119)(46,120)(47,101)(48,102)(49,103)(50,104)(51,105)(52,106)(53,107)(54,108)(55,109)(56,110)(57,111)(58,112)(59,113)(60,114)(81,149)(82,150)(83,151)(84,152)(85,153)(86,154)(87,155)(88,156)(89,157)(90,158)(91,159)(92,160)(93,141)(94,142)(95,143)(96,144)(97,145)(98,146)(99,147)(100,148)(121,231)(122,232)(123,233)(124,234)(125,235)(126,236)(127,237)(128,238)(129,239)(130,240)(131,221)(132,222)(133,223)(134,224)(135,225)(136,226)(137,227)(138,228)(139,229)(140,230)(161,192)(162,193)(163,194)(164,195)(165,196)(166,197)(167,198)(168,199)(169,200)(170,181)(171,182)(172,183)(173,184)(174,185)(175,186)(176,187)(177,188)(178,189)(179,190)(180,191), (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), (1,225,62,125)(2,124,63,224)(3,223,64,123)(4,122,65,222)(5,221,66,121)(6,140,67,240)(7,239,68,139)(8,138,69,238)(9,237,70,137)(10,136,71,236)(11,235,72,135)(12,134,73,234)(13,233,74,133)(14,132,75,232)(15,231,76,131)(16,130,77,230)(17,229,78,129)(18,128,79,228)(19,227,80,127)(20,126,61,226)(21,103,213,59)(22,58,214,102)(23,101,215,57)(24,56,216,120)(25,119,217,55)(26,54,218,118)(27,117,219,53)(28,52,220,116)(29,115,201,51)(30,50,202,114)(31,113,203,49)(32,48,204,112)(33,111,205,47)(34,46,206,110)(35,109,207,45)(36,44,208,108)(37,107,209,43)(38,42,210,106)(39,105,211,41)(40,60,212,104)(81,195,159,174)(82,173,160,194)(83,193,141,172)(84,171,142,192)(85,191,143,170)(86,169,144,190)(87,189,145,168)(88,167,146,188)(89,187,147,166)(90,165,148,186)(91,185,149,164)(92,163,150,184)(93,183,151,162)(94,161,152,182)(95,181,153,180)(96,179,154,200)(97,199,155,178)(98,177,156,198)(99,197,157,176)(100,175,158,196)>;
G:=Group( (1,29,182)(2,30,183)(3,31,184)(4,32,185)(5,33,186)(6,34,187)(7,35,188)(8,36,189)(9,37,190)(10,38,191)(11,39,192)(12,40,193)(13,21,194)(14,22,195)(15,23,196)(16,24,197)(17,25,198)(18,26,199)(19,27,200)(20,28,181)(41,142,135)(42,143,136)(43,144,137)(44,145,138)(45,146,139)(46,147,140)(47,148,121)(48,149,122)(49,150,123)(50,151,124)(51,152,125)(52,153,126)(53,154,127)(54,155,128)(55,156,129)(56,157,130)(57,158,131)(58,159,132)(59,160,133)(60,141,134)(61,220,180)(62,201,161)(63,202,162)(64,203,163)(65,204,164)(66,205,165)(67,206,166)(68,207,167)(69,208,168)(70,209,169)(71,210,170)(72,211,171)(73,212,172)(74,213,173)(75,214,174)(76,215,175)(77,216,176)(78,217,177)(79,218,178)(80,219,179)(81,232,102)(82,233,103)(83,234,104)(84,235,105)(85,236,106)(86,237,107)(87,238,108)(88,239,109)(89,240,110)(90,221,111)(91,222,112)(92,223,113)(93,224,114)(94,225,115)(95,226,116)(96,227,117)(97,228,118)(98,229,119)(99,230,120)(100,231,101), (41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(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)(221,231)(222,232)(223,233)(224,234)(225,235)(226,236)(227,237)(228,238)(229,239)(230,240), (1,72)(2,73)(3,74)(4,75)(5,76)(6,77)(7,78)(8,79)(9,80)(10,61)(11,62)(12,63)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,70)(20,71)(21,203)(22,204)(23,205)(24,206)(25,207)(26,208)(27,209)(28,210)(29,211)(30,212)(31,213)(32,214)(33,215)(34,216)(35,217)(36,218)(37,219)(38,220)(39,201)(40,202)(41,115)(42,116)(43,117)(44,118)(45,119)(46,120)(47,101)(48,102)(49,103)(50,104)(51,105)(52,106)(53,107)(54,108)(55,109)(56,110)(57,111)(58,112)(59,113)(60,114)(81,149)(82,150)(83,151)(84,152)(85,153)(86,154)(87,155)(88,156)(89,157)(90,158)(91,159)(92,160)(93,141)(94,142)(95,143)(96,144)(97,145)(98,146)(99,147)(100,148)(121,231)(122,232)(123,233)(124,234)(125,235)(126,236)(127,237)(128,238)(129,239)(130,240)(131,221)(132,222)(133,223)(134,224)(135,225)(136,226)(137,227)(138,228)(139,229)(140,230)(161,192)(162,193)(163,194)(164,195)(165,196)(166,197)(167,198)(168,199)(169,200)(170,181)(171,182)(172,183)(173,184)(174,185)(175,186)(176,187)(177,188)(178,189)(179,190)(180,191), (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), (1,225,62,125)(2,124,63,224)(3,223,64,123)(4,122,65,222)(5,221,66,121)(6,140,67,240)(7,239,68,139)(8,138,69,238)(9,237,70,137)(10,136,71,236)(11,235,72,135)(12,134,73,234)(13,233,74,133)(14,132,75,232)(15,231,76,131)(16,130,77,230)(17,229,78,129)(18,128,79,228)(19,227,80,127)(20,126,61,226)(21,103,213,59)(22,58,214,102)(23,101,215,57)(24,56,216,120)(25,119,217,55)(26,54,218,118)(27,117,219,53)(28,52,220,116)(29,115,201,51)(30,50,202,114)(31,113,203,49)(32,48,204,112)(33,111,205,47)(34,46,206,110)(35,109,207,45)(36,44,208,108)(37,107,209,43)(38,42,210,106)(39,105,211,41)(40,60,212,104)(81,195,159,174)(82,173,160,194)(83,193,141,172)(84,171,142,192)(85,191,143,170)(86,169,144,190)(87,189,145,168)(88,167,146,188)(89,187,147,166)(90,165,148,186)(91,185,149,164)(92,163,150,184)(93,183,151,162)(94,161,152,182)(95,181,153,180)(96,179,154,200)(97,199,155,178)(98,177,156,198)(99,197,157,176)(100,175,158,196) );
G=PermutationGroup([[(1,29,182),(2,30,183),(3,31,184),(4,32,185),(5,33,186),(6,34,187),(7,35,188),(8,36,189),(9,37,190),(10,38,191),(11,39,192),(12,40,193),(13,21,194),(14,22,195),(15,23,196),(16,24,197),(17,25,198),(18,26,199),(19,27,200),(20,28,181),(41,142,135),(42,143,136),(43,144,137),(44,145,138),(45,146,139),(46,147,140),(47,148,121),(48,149,122),(49,150,123),(50,151,124),(51,152,125),(52,153,126),(53,154,127),(54,155,128),(55,156,129),(56,157,130),(57,158,131),(58,159,132),(59,160,133),(60,141,134),(61,220,180),(62,201,161),(63,202,162),(64,203,163),(65,204,164),(66,205,165),(67,206,166),(68,207,167),(69,208,168),(70,209,169),(71,210,170),(72,211,171),(73,212,172),(74,213,173),(75,214,174),(76,215,175),(77,216,176),(78,217,177),(79,218,178),(80,219,179),(81,232,102),(82,233,103),(83,234,104),(84,235,105),(85,236,106),(86,237,107),(87,238,108),(88,239,109),(89,240,110),(90,221,111),(91,222,112),(92,223,113),(93,224,114),(94,225,115),(95,226,116),(96,227,117),(97,228,118),(98,229,119),(99,230,120),(100,231,101)], [(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(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),(221,231),(222,232),(223,233),(224,234),(225,235),(226,236),(227,237),(228,238),(229,239),(230,240)], [(1,72),(2,73),(3,74),(4,75),(5,76),(6,77),(7,78),(8,79),(9,80),(10,61),(11,62),(12,63),(13,64),(14,65),(15,66),(16,67),(17,68),(18,69),(19,70),(20,71),(21,203),(22,204),(23,205),(24,206),(25,207),(26,208),(27,209),(28,210),(29,211),(30,212),(31,213),(32,214),(33,215),(34,216),(35,217),(36,218),(37,219),(38,220),(39,201),(40,202),(41,115),(42,116),(43,117),(44,118),(45,119),(46,120),(47,101),(48,102),(49,103),(50,104),(51,105),(52,106),(53,107),(54,108),(55,109),(56,110),(57,111),(58,112),(59,113),(60,114),(81,149),(82,150),(83,151),(84,152),(85,153),(86,154),(87,155),(88,156),(89,157),(90,158),(91,159),(92,160),(93,141),(94,142),(95,143),(96,144),(97,145),(98,146),(99,147),(100,148),(121,231),(122,232),(123,233),(124,234),(125,235),(126,236),(127,237),(128,238),(129,239),(130,240),(131,221),(132,222),(133,223),(134,224),(135,225),(136,226),(137,227),(138,228),(139,229),(140,230),(161,192),(162,193),(163,194),(164,195),(165,196),(166,197),(167,198),(168,199),(169,200),(170,181),(171,182),(172,183),(173,184),(174,185),(175,186),(176,187),(177,188),(178,189),(179,190),(180,191)], [(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)], [(1,225,62,125),(2,124,63,224),(3,223,64,123),(4,122,65,222),(5,221,66,121),(6,140,67,240),(7,239,68,139),(8,138,69,238),(9,237,70,137),(10,136,71,236),(11,235,72,135),(12,134,73,234),(13,233,74,133),(14,132,75,232),(15,231,76,131),(16,130,77,230),(17,229,78,129),(18,128,79,228),(19,227,80,127),(20,126,61,226),(21,103,213,59),(22,58,214,102),(23,101,215,57),(24,56,216,120),(25,119,217,55),(26,54,218,118),(27,117,219,53),(28,52,220,116),(29,115,201,51),(30,50,202,114),(31,113,203,49),(32,48,204,112),(33,111,205,47),(34,46,206,110),(35,109,207,45),(36,44,208,108),(37,107,209,43),(38,42,210,106),(39,105,211,41),(40,60,212,104),(81,195,159,174),(82,173,160,194),(83,193,141,172),(84,171,142,192),(85,191,143,170),(86,169,144,190),(87,189,145,168),(88,167,146,188),(89,187,147,166),(90,165,148,186),(91,185,149,164),(92,163,150,184),(93,183,151,162),(94,161,152,182),(95,181,153,180),(96,179,154,200),(97,199,155,178),(98,177,156,198),(99,197,157,176),(100,175,158,196)]])
138 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 | ··· | 10N | 12A | ··· | 12H | 12I | ··· | 12N | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 30A | ··· | 30AB | 60A | ··· | 60AF |
| 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 | 12 | ··· | 12 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 1 | 1 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 20 | 20 | 2 | ··· | 2 | 2 | ··· | 2 | 20 | ··· | 20 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
138 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 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | D4 | D5 | C4○D4 | D10 | D10 | C3×D4 | C3×D5 | C5⋊D4 | C3×C4○D4 | C6×D5 | C6×D5 | C4○D20 | C3×C5⋊D4 | C3×C4○D20 |
| kernel | C3×C23.23D10 | C3×C10.D4 | C3×D10⋊C4 | C3×C23.D5 | C6×C5⋊D4 | C22×C60 | C23.23D10 | C10.D4 | D10⋊C4 | C23.D5 | C2×C5⋊D4 | C22×C20 | C2×C30 | C22×C12 | C30 | C2×C12 | C22×C6 | C2×C10 | C22×C4 | C2×C6 | C10 | C2×C4 | C23 | C6 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 4 | 4 | 8 | 8 | 8 | 4 | 16 | 16 | 32 |
Matrix representation of C3×C23.23D10 ►in GL4(𝔽61) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 1 | 0 | 0 | 0 |
| 46 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 8 | 0 | 0 | 0 |
| 42 | 38 | 0 | 0 |
| 0 | 0 | 39 | 53 |
| 0 | 0 | 55 | 45 |
| 36 | 17 | 0 | 0 |
| 17 | 25 | 0 | 0 |
| 0 | 0 | 3 | 52 |
| 0 | 0 | 35 | 58 |
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,13,0,0,0,0,13],[1,46,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[8,42,0,0,0,38,0,0,0,0,39,55,0,0,53,45],[36,17,0,0,17,25,0,0,0,0,3,35,0,0,52,58] >;
C3×C23.23D10 in GAP, Magma, Sage, TeX
C_3\times C_2^3._{23}D_{10} % in TeX
G:=Group("C3xC2^3.23D10"); // GroupNames label
G:=SmallGroup(480,722);
// by ID
G=gap.SmallGroup(480,722);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,701,590,268,18822]);
// 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=c*e^9>;
// generators/relations