direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C23.11D10, (C4×Dic5)⋊9C6, (C2×Dic5)⋊4C12, (C6×Dic5)⋊10C4, C10.D4⋊7C6, C23.15(C6×D5), C22.6(D5×C12), C23.D5.1C6, (C12×Dic5)⋊27C2, (C2×C12).270D10, (C22×C6).71D10, C15⋊25(C42⋊C2), C30.227(C4○D4), C10.18(C22×C12), (C2×C30).335C23, C30.176(C22×C4), (C2×C60).393C22, Dic5.20(C2×C12), C6.106(D4⋊2D5), (C22×Dic5).4C6, (C22×C30).93C22, (C6×Dic5).282C22, C2.7(D5×C2×C12), C6.101(C2×C4×D5), C5⋊3(C3×C42⋊C2), (C2×C6).40(C4×D5), (C2×C4).21(C6×D5), C22.12(D5×C2×C6), (C2×C20).47(C2×C6), C10.18(C3×C4○D4), C2.1(C3×D4⋊2D5), (C5×C22⋊C4).3C6, C22⋊C4.3(C3×D5), (C3×C22⋊C4).6D5, (C2×C6×Dic5).10C2, (C2×C30).121(C2×C4), (C2×C10).24(C2×C12), (C15×C22⋊C4).8C2, (C3×C23.D5).7C2, (C3×C10.D4)⋊23C2, (C22×C10).12(C2×C6), (C2×C10).18(C22×C6), (C2×Dic5).26(C2×C6), (C3×Dic5).62(C2×C4), (C2×C6).331(C22×D5), SmallGroup(480,670)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C23.11D10
G = < a,b,c,d,e,f | a3=b2=c2=d2=1, e10=c, 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=e9 >
Subgroups: 368 in 152 conjugacy classes, 82 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, C23, C10, C10, C10, C12, C2×C6, C2×C6, C2×C6, C15, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, Dic5, Dic5, C20, C2×C10, C2×C10, C2×C10, C2×C12, C2×C12, C22×C6, C30, C30, C30, C42⋊C2, C2×Dic5, C2×Dic5, C2×C20, C22×C10, C4×C12, C3×C22⋊C4, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C3×Dic5, C3×Dic5, C60, C2×C30, C2×C30, C2×C30, C4×Dic5, C10.D4, C23.D5, C5×C22⋊C4, C22×Dic5, C3×C42⋊C2, C6×Dic5, C6×Dic5, C2×C60, C22×C30, C23.11D10, C12×Dic5, C3×C10.D4, C3×C23.D5, C15×C22⋊C4, C2×C6×Dic5, C3×C23.11D10
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, D5, C12, C2×C6, C22×C4, C4○D4, D10, C2×C12, C22×C6, C3×D5, C42⋊C2, C4×D5, C22×D5, C22×C12, C3×C4○D4, C6×D5, C2×C4×D5, D4⋊2D5, C3×C42⋊C2, D5×C12, D5×C2×C6, C23.11D10, D5×C2×C12, C3×D4⋊2D5, C3×C23.11D10
►On 240 points
Generators in S240
(1 183 153)(2 184 154)(3 185 155)(4 186 156)(5 187 157)(6 188 158)(7 189 159)(8 190 160)(9 191 141)(10 192 142)(11 193 143)(12 194 144)(13 195 145)(14 196 146)(15 197 147)(16 198 148)(17 199 149)(18 200 150)(19 181 151)(20 182 152)(21 97 124)(22 98 125)(23 99 126)(24 100 127)(25 81 128)(26 82 129)(27 83 130)(28 84 131)(29 85 132)(30 86 133)(31 87 134)(32 88 135)(33 89 136)(34 90 137)(35 91 138)(36 92 139)(37 93 140)(38 94 121)(39 95 122)(40 96 123)(41 218 233)(42 219 234)(43 220 235)(44 201 236)(45 202 237)(46 203 238)(47 204 239)(48 205 240)(49 206 221)(50 207 222)(51 208 223)(52 209 224)(53 210 225)(54 211 226)(55 212 227)(56 213 228)(57 214 229)(58 215 230)(59 216 231)(60 217 232)(61 117 170)(62 118 171)(63 119 172)(64 120 173)(65 101 174)(66 102 175)(67 103 176)(68 104 177)(69 105 178)(70 106 179)(71 107 180)(72 108 161)(73 109 162)(74 110 163)(75 111 164)(76 112 165)(77 113 166)(78 114 167)(79 115 168)(80 116 169)
(1 11)(2 131)(3 13)(4 133)(5 15)(6 135)(7 17)(8 137)(9 19)(10 139)(12 121)(14 123)(16 125)(18 127)(20 129)(21 31)(22 198)(23 33)(24 200)(25 35)(26 182)(27 37)(28 184)(29 39)(30 186)(32 188)(34 190)(36 192)(38 194)(40 196)(41 51)(42 163)(43 53)(44 165)(45 55)(46 167)(47 57)(48 169)(49 59)(50 171)(52 173)(54 175)(56 177)(58 179)(60 161)(61 71)(62 207)(63 73)(64 209)(65 75)(66 211)(67 77)(68 213)(69 79)(70 215)(72 217)(74 219)(76 201)(78 203)(80 205)(81 91)(82 152)(83 93)(84 154)(85 95)(86 156)(87 97)(88 158)(89 99)(90 160)(92 142)(94 144)(96 146)(98 148)(100 150)(101 111)(102 226)(103 113)(104 228)(105 115)(106 230)(107 117)(108 232)(109 119)(110 234)(112 236)(114 238)(116 240)(118 222)(120 224)(122 132)(124 134)(126 136)(128 138)(130 140)(141 151)(143 153)(145 155)(147 157)(149 159)(162 172)(164 174)(166 176)(168 178)(170 180)(181 191)(183 193)(185 195)(187 197)(189 199)(202 212)(204 214)(206 216)(208 218)(210 220)(221 231)(223 233)(225 235)(227 237)(229 239)
(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)
(1 140)(2 121)(3 122)(4 123)(5 124)(6 125)(7 126)(8 127)(9 128)(10 129)(11 130)(12 131)(13 132)(14 133)(15 134)(16 135)(17 136)(18 137)(19 138)(20 139)(21 187)(22 188)(23 189)(24 190)(25 191)(26 192)(27 193)(28 194)(29 195)(30 196)(31 197)(32 198)(33 199)(34 200)(35 181)(36 182)(37 183)(38 184)(39 185)(40 186)(41 172)(42 173)(43 174)(44 175)(45 176)(46 177)(47 178)(48 179)(49 180)(50 161)(51 162)(52 163)(53 164)(54 165)(55 166)(56 167)(57 168)(58 169)(59 170)(60 171)(61 216)(62 217)(63 218)(64 219)(65 220)(66 201)(67 202)(68 203)(69 204)(70 205)(71 206)(72 207)(73 208)(74 209)(75 210)(76 211)(77 212)(78 213)(79 214)(80 215)(81 141)(82 142)(83 143)(84 144)(85 145)(86 146)(87 147)(88 148)(89 149)(90 150)(91 151)(92 152)(93 153)(94 154)(95 155)(96 156)(97 157)(98 158)(99 159)(100 160)(101 235)(102 236)(103 237)(104 238)(105 239)(106 240)(107 221)(108 222)(109 223)(110 224)(111 225)(112 226)(113 227)(114 228)(115 229)(116 230)(117 231)(118 232)(119 233)(120 234)
(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 54 130 175)(2 43 131 164)(3 52 132 173)(4 41 133 162)(5 50 134 171)(6 59 135 180)(7 48 136 169)(8 57 137 178)(9 46 138 167)(10 55 139 176)(11 44 140 165)(12 53 121 174)(13 42 122 163)(14 51 123 172)(15 60 124 161)(16 49 125 170)(17 58 126 179)(18 47 127 168)(19 56 128 177)(20 45 129 166)(21 72 197 217)(22 61 198 206)(23 70 199 215)(24 79 200 204)(25 68 181 213)(26 77 182 202)(27 66 183 211)(28 75 184 220)(29 64 185 209)(30 73 186 218)(31 62 187 207)(32 71 188 216)(33 80 189 205)(34 69 190 214)(35 78 191 203)(36 67 192 212)(37 76 193 201)(38 65 194 210)(39 74 195 219)(40 63 196 208)(81 104 151 228)(82 113 152 237)(83 102 153 226)(84 111 154 235)(85 120 155 224)(86 109 156 233)(87 118 157 222)(88 107 158 231)(89 116 159 240)(90 105 160 229)(91 114 141 238)(92 103 142 227)(93 112 143 236)(94 101 144 225)(95 110 145 234)(96 119 146 223)(97 108 147 232)(98 117 148 221)(99 106 149 230)(100 115 150 239)
G:=sub<Sym(240)| (1,183,153)(2,184,154)(3,185,155)(4,186,156)(5,187,157)(6,188,158)(7,189,159)(8,190,160)(9,191,141)(10,192,142)(11,193,143)(12,194,144)(13,195,145)(14,196,146)(15,197,147)(16,198,148)(17,199,149)(18,200,150)(19,181,151)(20,182,152)(21,97,124)(22,98,125)(23,99,126)(24,100,127)(25,81,128)(26,82,129)(27,83,130)(28,84,131)(29,85,132)(30,86,133)(31,87,134)(32,88,135)(33,89,136)(34,90,137)(35,91,138)(36,92,139)(37,93,140)(38,94,121)(39,95,122)(40,96,123)(41,218,233)(42,219,234)(43,220,235)(44,201,236)(45,202,237)(46,203,238)(47,204,239)(48,205,240)(49,206,221)(50,207,222)(51,208,223)(52,209,224)(53,210,225)(54,211,226)(55,212,227)(56,213,228)(57,214,229)(58,215,230)(59,216,231)(60,217,232)(61,117,170)(62,118,171)(63,119,172)(64,120,173)(65,101,174)(66,102,175)(67,103,176)(68,104,177)(69,105,178)(70,106,179)(71,107,180)(72,108,161)(73,109,162)(74,110,163)(75,111,164)(76,112,165)(77,113,166)(78,114,167)(79,115,168)(80,116,169), (1,11)(2,131)(3,13)(4,133)(5,15)(6,135)(7,17)(8,137)(9,19)(10,139)(12,121)(14,123)(16,125)(18,127)(20,129)(21,31)(22,198)(23,33)(24,200)(25,35)(26,182)(27,37)(28,184)(29,39)(30,186)(32,188)(34,190)(36,192)(38,194)(40,196)(41,51)(42,163)(43,53)(44,165)(45,55)(46,167)(47,57)(48,169)(49,59)(50,171)(52,173)(54,175)(56,177)(58,179)(60,161)(61,71)(62,207)(63,73)(64,209)(65,75)(66,211)(67,77)(68,213)(69,79)(70,215)(72,217)(74,219)(76,201)(78,203)(80,205)(81,91)(82,152)(83,93)(84,154)(85,95)(86,156)(87,97)(88,158)(89,99)(90,160)(92,142)(94,144)(96,146)(98,148)(100,150)(101,111)(102,226)(103,113)(104,228)(105,115)(106,230)(107,117)(108,232)(109,119)(110,234)(112,236)(114,238)(116,240)(118,222)(120,224)(122,132)(124,134)(126,136)(128,138)(130,140)(141,151)(143,153)(145,155)(147,157)(149,159)(162,172)(164,174)(166,176)(168,178)(170,180)(181,191)(183,193)(185,195)(187,197)(189,199)(202,212)(204,214)(206,216)(208,218)(210,220)(221,231)(223,233)(225,235)(227,237)(229,239), (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), (1,140)(2,121)(3,122)(4,123)(5,124)(6,125)(7,126)(8,127)(9,128)(10,129)(11,130)(12,131)(13,132)(14,133)(15,134)(16,135)(17,136)(18,137)(19,138)(20,139)(21,187)(22,188)(23,189)(24,190)(25,191)(26,192)(27,193)(28,194)(29,195)(30,196)(31,197)(32,198)(33,199)(34,200)(35,181)(36,182)(37,183)(38,184)(39,185)(40,186)(41,172)(42,173)(43,174)(44,175)(45,176)(46,177)(47,178)(48,179)(49,180)(50,161)(51,162)(52,163)(53,164)(54,165)(55,166)(56,167)(57,168)(58,169)(59,170)(60,171)(61,216)(62,217)(63,218)(64,219)(65,220)(66,201)(67,202)(68,203)(69,204)(70,205)(71,206)(72,207)(73,208)(74,209)(75,210)(76,211)(77,212)(78,213)(79,214)(80,215)(81,141)(82,142)(83,143)(84,144)(85,145)(86,146)(87,147)(88,148)(89,149)(90,150)(91,151)(92,152)(93,153)(94,154)(95,155)(96,156)(97,157)(98,158)(99,159)(100,160)(101,235)(102,236)(103,237)(104,238)(105,239)(106,240)(107,221)(108,222)(109,223)(110,224)(111,225)(112,226)(113,227)(114,228)(115,229)(116,230)(117,231)(118,232)(119,233)(120,234), (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), (1,54,130,175)(2,43,131,164)(3,52,132,173)(4,41,133,162)(5,50,134,171)(6,59,135,180)(7,48,136,169)(8,57,137,178)(9,46,138,167)(10,55,139,176)(11,44,140,165)(12,53,121,174)(13,42,122,163)(14,51,123,172)(15,60,124,161)(16,49,125,170)(17,58,126,179)(18,47,127,168)(19,56,128,177)(20,45,129,166)(21,72,197,217)(22,61,198,206)(23,70,199,215)(24,79,200,204)(25,68,181,213)(26,77,182,202)(27,66,183,211)(28,75,184,220)(29,64,185,209)(30,73,186,218)(31,62,187,207)(32,71,188,216)(33,80,189,205)(34,69,190,214)(35,78,191,203)(36,67,192,212)(37,76,193,201)(38,65,194,210)(39,74,195,219)(40,63,196,208)(81,104,151,228)(82,113,152,237)(83,102,153,226)(84,111,154,235)(85,120,155,224)(86,109,156,233)(87,118,157,222)(88,107,158,231)(89,116,159,240)(90,105,160,229)(91,114,141,238)(92,103,142,227)(93,112,143,236)(94,101,144,225)(95,110,145,234)(96,119,146,223)(97,108,147,232)(98,117,148,221)(99,106,149,230)(100,115,150,239)>;
G:=Group( (1,183,153)(2,184,154)(3,185,155)(4,186,156)(5,187,157)(6,188,158)(7,189,159)(8,190,160)(9,191,141)(10,192,142)(11,193,143)(12,194,144)(13,195,145)(14,196,146)(15,197,147)(16,198,148)(17,199,149)(18,200,150)(19,181,151)(20,182,152)(21,97,124)(22,98,125)(23,99,126)(24,100,127)(25,81,128)(26,82,129)(27,83,130)(28,84,131)(29,85,132)(30,86,133)(31,87,134)(32,88,135)(33,89,136)(34,90,137)(35,91,138)(36,92,139)(37,93,140)(38,94,121)(39,95,122)(40,96,123)(41,218,233)(42,219,234)(43,220,235)(44,201,236)(45,202,237)(46,203,238)(47,204,239)(48,205,240)(49,206,221)(50,207,222)(51,208,223)(52,209,224)(53,210,225)(54,211,226)(55,212,227)(56,213,228)(57,214,229)(58,215,230)(59,216,231)(60,217,232)(61,117,170)(62,118,171)(63,119,172)(64,120,173)(65,101,174)(66,102,175)(67,103,176)(68,104,177)(69,105,178)(70,106,179)(71,107,180)(72,108,161)(73,109,162)(74,110,163)(75,111,164)(76,112,165)(77,113,166)(78,114,167)(79,115,168)(80,116,169), (1,11)(2,131)(3,13)(4,133)(5,15)(6,135)(7,17)(8,137)(9,19)(10,139)(12,121)(14,123)(16,125)(18,127)(20,129)(21,31)(22,198)(23,33)(24,200)(25,35)(26,182)(27,37)(28,184)(29,39)(30,186)(32,188)(34,190)(36,192)(38,194)(40,196)(41,51)(42,163)(43,53)(44,165)(45,55)(46,167)(47,57)(48,169)(49,59)(50,171)(52,173)(54,175)(56,177)(58,179)(60,161)(61,71)(62,207)(63,73)(64,209)(65,75)(66,211)(67,77)(68,213)(69,79)(70,215)(72,217)(74,219)(76,201)(78,203)(80,205)(81,91)(82,152)(83,93)(84,154)(85,95)(86,156)(87,97)(88,158)(89,99)(90,160)(92,142)(94,144)(96,146)(98,148)(100,150)(101,111)(102,226)(103,113)(104,228)(105,115)(106,230)(107,117)(108,232)(109,119)(110,234)(112,236)(114,238)(116,240)(118,222)(120,224)(122,132)(124,134)(126,136)(128,138)(130,140)(141,151)(143,153)(145,155)(147,157)(149,159)(162,172)(164,174)(166,176)(168,178)(170,180)(181,191)(183,193)(185,195)(187,197)(189,199)(202,212)(204,214)(206,216)(208,218)(210,220)(221,231)(223,233)(225,235)(227,237)(229,239), (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), (1,140)(2,121)(3,122)(4,123)(5,124)(6,125)(7,126)(8,127)(9,128)(10,129)(11,130)(12,131)(13,132)(14,133)(15,134)(16,135)(17,136)(18,137)(19,138)(20,139)(21,187)(22,188)(23,189)(24,190)(25,191)(26,192)(27,193)(28,194)(29,195)(30,196)(31,197)(32,198)(33,199)(34,200)(35,181)(36,182)(37,183)(38,184)(39,185)(40,186)(41,172)(42,173)(43,174)(44,175)(45,176)(46,177)(47,178)(48,179)(49,180)(50,161)(51,162)(52,163)(53,164)(54,165)(55,166)(56,167)(57,168)(58,169)(59,170)(60,171)(61,216)(62,217)(63,218)(64,219)(65,220)(66,201)(67,202)(68,203)(69,204)(70,205)(71,206)(72,207)(73,208)(74,209)(75,210)(76,211)(77,212)(78,213)(79,214)(80,215)(81,141)(82,142)(83,143)(84,144)(85,145)(86,146)(87,147)(88,148)(89,149)(90,150)(91,151)(92,152)(93,153)(94,154)(95,155)(96,156)(97,157)(98,158)(99,159)(100,160)(101,235)(102,236)(103,237)(104,238)(105,239)(106,240)(107,221)(108,222)(109,223)(110,224)(111,225)(112,226)(113,227)(114,228)(115,229)(116,230)(117,231)(118,232)(119,233)(120,234), (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), (1,54,130,175)(2,43,131,164)(3,52,132,173)(4,41,133,162)(5,50,134,171)(6,59,135,180)(7,48,136,169)(8,57,137,178)(9,46,138,167)(10,55,139,176)(11,44,140,165)(12,53,121,174)(13,42,122,163)(14,51,123,172)(15,60,124,161)(16,49,125,170)(17,58,126,179)(18,47,127,168)(19,56,128,177)(20,45,129,166)(21,72,197,217)(22,61,198,206)(23,70,199,215)(24,79,200,204)(25,68,181,213)(26,77,182,202)(27,66,183,211)(28,75,184,220)(29,64,185,209)(30,73,186,218)(31,62,187,207)(32,71,188,216)(33,80,189,205)(34,69,190,214)(35,78,191,203)(36,67,192,212)(37,76,193,201)(38,65,194,210)(39,74,195,219)(40,63,196,208)(81,104,151,228)(82,113,152,237)(83,102,153,226)(84,111,154,235)(85,120,155,224)(86,109,156,233)(87,118,157,222)(88,107,158,231)(89,116,159,240)(90,105,160,229)(91,114,141,238)(92,103,142,227)(93,112,143,236)(94,101,144,225)(95,110,145,234)(96,119,146,223)(97,108,147,232)(98,117,148,221)(99,106,149,230)(100,115,150,239) );
G=PermutationGroup([[(1,183,153),(2,184,154),(3,185,155),(4,186,156),(5,187,157),(6,188,158),(7,189,159),(8,190,160),(9,191,141),(10,192,142),(11,193,143),(12,194,144),(13,195,145),(14,196,146),(15,197,147),(16,198,148),(17,199,149),(18,200,150),(19,181,151),(20,182,152),(21,97,124),(22,98,125),(23,99,126),(24,100,127),(25,81,128),(26,82,129),(27,83,130),(28,84,131),(29,85,132),(30,86,133),(31,87,134),(32,88,135),(33,89,136),(34,90,137),(35,91,138),(36,92,139),(37,93,140),(38,94,121),(39,95,122),(40,96,123),(41,218,233),(42,219,234),(43,220,235),(44,201,236),(45,202,237),(46,203,238),(47,204,239),(48,205,240),(49,206,221),(50,207,222),(51,208,223),(52,209,224),(53,210,225),(54,211,226),(55,212,227),(56,213,228),(57,214,229),(58,215,230),(59,216,231),(60,217,232),(61,117,170),(62,118,171),(63,119,172),(64,120,173),(65,101,174),(66,102,175),(67,103,176),(68,104,177),(69,105,178),(70,106,179),(71,107,180),(72,108,161),(73,109,162),(74,110,163),(75,111,164),(76,112,165),(77,113,166),(78,114,167),(79,115,168),(80,116,169)], [(1,11),(2,131),(3,13),(4,133),(5,15),(6,135),(7,17),(8,137),(9,19),(10,139),(12,121),(14,123),(16,125),(18,127),(20,129),(21,31),(22,198),(23,33),(24,200),(25,35),(26,182),(27,37),(28,184),(29,39),(30,186),(32,188),(34,190),(36,192),(38,194),(40,196),(41,51),(42,163),(43,53),(44,165),(45,55),(46,167),(47,57),(48,169),(49,59),(50,171),(52,173),(54,175),(56,177),(58,179),(60,161),(61,71),(62,207),(63,73),(64,209),(65,75),(66,211),(67,77),(68,213),(69,79),(70,215),(72,217),(74,219),(76,201),(78,203),(80,205),(81,91),(82,152),(83,93),(84,154),(85,95),(86,156),(87,97),(88,158),(89,99),(90,160),(92,142),(94,144),(96,146),(98,148),(100,150),(101,111),(102,226),(103,113),(104,228),(105,115),(106,230),(107,117),(108,232),(109,119),(110,234),(112,236),(114,238),(116,240),(118,222),(120,224),(122,132),(124,134),(126,136),(128,138),(130,140),(141,151),(143,153),(145,155),(147,157),(149,159),(162,172),(164,174),(166,176),(168,178),(170,180),(181,191),(183,193),(185,195),(187,197),(189,199),(202,212),(204,214),(206,216),(208,218),(210,220),(221,231),(223,233),(225,235),(227,237),(229,239)], [(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)], [(1,140),(2,121),(3,122),(4,123),(5,124),(6,125),(7,126),(8,127),(9,128),(10,129),(11,130),(12,131),(13,132),(14,133),(15,134),(16,135),(17,136),(18,137),(19,138),(20,139),(21,187),(22,188),(23,189),(24,190),(25,191),(26,192),(27,193),(28,194),(29,195),(30,196),(31,197),(32,198),(33,199),(34,200),(35,181),(36,182),(37,183),(38,184),(39,185),(40,186),(41,172),(42,173),(43,174),(44,175),(45,176),(46,177),(47,178),(48,179),(49,180),(50,161),(51,162),(52,163),(53,164),(54,165),(55,166),(56,167),(57,168),(58,169),(59,170),(60,171),(61,216),(62,217),(63,218),(64,219),(65,220),(66,201),(67,202),(68,203),(69,204),(70,205),(71,206),(72,207),(73,208),(74,209),(75,210),(76,211),(77,212),(78,213),(79,214),(80,215),(81,141),(82,142),(83,143),(84,144),(85,145),(86,146),(87,147),(88,148),(89,149),(90,150),(91,151),(92,152),(93,153),(94,154),(95,155),(96,156),(97,157),(98,158),(99,159),(100,160),(101,235),(102,236),(103,237),(104,238),(105,239),(106,240),(107,221),(108,222),(109,223),(110,224),(111,225),(112,226),(113,227),(114,228),(115,229),(116,230),(117,231),(118,232),(119,233),(120,234)], [(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)], [(1,54,130,175),(2,43,131,164),(3,52,132,173),(4,41,133,162),(5,50,134,171),(6,59,135,180),(7,48,136,169),(8,57,137,178),(9,46,138,167),(10,55,139,176),(11,44,140,165),(12,53,121,174),(13,42,122,163),(14,51,123,172),(15,60,124,161),(16,49,125,170),(17,58,126,179),(18,47,127,168),(19,56,128,177),(20,45,129,166),(21,72,197,217),(22,61,198,206),(23,70,199,215),(24,79,200,204),(25,68,181,213),(26,77,182,202),(27,66,183,211),(28,75,184,220),(29,64,185,209),(30,73,186,218),(31,62,187,207),(32,71,188,216),(33,80,189,205),(34,69,190,214),(35,78,191,203),(36,67,192,212),(37,76,193,201),(38,65,194,210),(39,74,195,219),(40,63,196,208),(81,104,151,228),(82,113,152,237),(83,102,153,226),(84,111,154,235),(85,120,155,224),(86,109,156,233),(87,118,157,222),(88,107,158,231),(89,116,159,240),(90,105,160,229),(91,114,141,238),(92,103,142,227),(93,112,143,236),(94,101,144,225),(95,110,145,234),(96,119,146,223),(97,108,147,232),(98,117,148,221),(99,106,149,230),(100,115,150,239)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 5A | 5B | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | ··· | 12H | 12I | ··· | 12P | 12Q | ··· | 12AB | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 30A | ··· | 30L | 30M | ··· | 30T | 60A | ··· | 60P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 5 | ··· | 5 | 10 | ··· | 10 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C6 | C12 | D5 | C4○D4 | D10 | D10 | C3×D5 | C4×D5 | C3×C4○D4 | C6×D5 | C6×D5 | D5×C12 | D4⋊2D5 | C3×D4⋊2D5 |
| kernel | C3×C23.11D10 | C12×Dic5 | C3×C10.D4 | C3×C23.D5 | C15×C22⋊C4 | C2×C6×Dic5 | C23.11D10 | C6×Dic5 | C4×Dic5 | C10.D4 | C23.D5 | C5×C22⋊C4 | C22×Dic5 | C2×Dic5 | C3×C22⋊C4 | C30 | C2×C12 | C22×C6 | C22⋊C4 | C2×C6 | C10 | C2×C4 | C23 | C22 | C6 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 8 | 4 | 4 | 2 | 2 | 2 | 16 | 2 | 4 | 4 | 2 | 4 | 8 | 8 | 8 | 4 | 16 | 4 | 8 |
Matrix representation of C3×C23.11D10 ►in GL4(𝔽61) generated by
| 47 | 0 | 0 | 0 |
| 0 | 47 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 60 | 60 |
| 60 | 0 | 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 |
| 57 | 4 | 0 | 0 |
| 46 | 0 | 0 | 0 |
| 0 | 0 | 60 | 59 |
| 0 | 0 | 0 | 1 |
| 34 | 37 | 0 | 0 |
| 5 | 27 | 0 | 0 |
| 0 | 0 | 11 | 22 |
| 0 | 0 | 0 | 50 |
G:=sub<GL(4,GF(61))| [47,0,0,0,0,47,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,1,60,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[57,46,0,0,4,0,0,0,0,0,60,0,0,0,59,1],[34,5,0,0,37,27,0,0,0,0,11,0,0,0,22,50] >;
C3×C23.11D10 in GAP, Magma, Sage, TeX
C_3\times C_2^3._{11}D_{10} % in TeX
G:=Group("C3xC2^3.11D10"); // GroupNames label
G:=SmallGroup(480,670);
// by ID
G=gap.SmallGroup(480,670);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,336,555,142,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=1,e^10=c,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=e^9>;
// generators/relations