direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C20.C23, C60.129D4, C60.204C23, Q8⋊D5⋊5C6, (C6×Q8)⋊9D5, (Q8×C10)⋊6C6, (Q8×C30)⋊9C2, C5⋊Q16⋊5C6, C4○D20.5C6, C20.19(C3×D4), C10.54(C6×D4), C4.Dic5⋊7C6, Q8.11(C6×D5), D20.10(C2×C6), (C2×C30).167D4, C30.411(C2×D4), (C3×Q8).38D10, (C2×C12).242D10, C12.96(C5⋊D4), C15⋊35(C8.C22), C20.15(C22×C6), Dic10.9(C2×C6), (C2×C60).297C22, (C3×D20).49C22, C12.204(C22×D5), (Q8×C15).43C22, (C3×Dic10).51C22, C4.15(D5×C2×C6), (C2×Q8)⋊4(C3×D5), C5⋊4(C3×C8.C22), (C3×Q8⋊D5)⋊13C2, C5⋊2C8.3(C2×C6), (C2×C4).16(C6×D5), C2.18(C6×C5⋊D4), C4.17(C3×C5⋊D4), (C2×C20).34(C2×C6), (C3×C5⋊Q16)⋊13C2, (C2×C10).42(C3×D4), C6.139(C2×C5⋊D4), (C5×Q8).14(C2×C6), (C3×C4○D20).11C2, (C2×C6).64(C5⋊D4), (C3×C4.Dic5)⋊19C2, C22.11(C3×C5⋊D4), (C3×C5⋊2C8).33C22, SmallGroup(480,735)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C20.C23
G = < a,b,c,d,e | a3=b20=c2=1, d2=e2=b10, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe-1=b11, cd=dc, ece-1=b5c, ede-1=b10d >
Subgroups: 352 in 120 conjugacy classes, 58 normal (38 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C24, C2×C12, C2×C12, C3×D4, C3×Q8, C3×Q8, C3×D5, C30, C30, C8.C22, C5⋊2C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20, C5×Q8, C5×Q8, C3×M4(2), C3×SD16, C3×Q16, C6×Q8, C3×C4○D4, C3×Dic5, C60, C60, C6×D5, C2×C30, C4.Dic5, Q8⋊D5, C5⋊Q16, C4○D20, Q8×C10, C3×C8.C22, C3×C5⋊2C8, C3×Dic10, D5×C12, C3×D20, C3×C5⋊D4, C2×C60, C2×C60, Q8×C15, Q8×C15, C20.C23, C3×C4.Dic5, C3×Q8⋊D5, C3×C5⋊Q16, C3×C4○D20, Q8×C30, C3×C20.C23
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, D10, C3×D4, C22×C6, C3×D5, C8.C22, C5⋊D4, C22×D5, C6×D4, C6×D5, C2×C5⋊D4, C3×C8.C22, C3×C5⋊D4, D5×C2×C6, C20.C23, C6×C5⋊D4, C3×C20.C23
►On 240 points
Generators in S240
(1 56 29)(2 57 30)(3 58 31)(4 59 32)(5 60 33)(6 41 34)(7 42 35)(8 43 36)(9 44 37)(10 45 38)(11 46 39)(12 47 40)(13 48 21)(14 49 22)(15 50 23)(16 51 24)(17 52 25)(18 53 26)(19 54 27)(20 55 28)(61 118 89)(62 119 90)(63 120 91)(64 101 92)(65 102 93)(66 103 94)(67 104 95)(68 105 96)(69 106 97)(70 107 98)(71 108 99)(72 109 100)(73 110 81)(74 111 82)(75 112 83)(76 113 84)(77 114 85)(78 115 86)(79 116 87)(80 117 88)(121 166 151)(122 167 152)(123 168 153)(124 169 154)(125 170 155)(126 171 156)(127 172 157)(128 173 158)(129 174 159)(130 175 160)(131 176 141)(132 177 142)(133 178 143)(134 179 144)(135 180 145)(136 161 146)(137 162 147)(138 163 148)(139 164 149)(140 165 150)(181 231 211)(182 232 212)(183 233 213)(184 234 214)(185 235 215)(186 236 216)(187 237 217)(188 238 218)(189 239 219)(190 240 220)(191 221 201)(192 222 202)(193 223 203)(194 224 204)(195 225 205)(196 226 206)(197 227 207)(198 228 208)(199 229 209)(200 230 210)
(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)
(1 121)(2 140)(3 139)(4 138)(5 137)(6 136)(7 135)(8 134)(9 133)(10 132)(11 131)(12 130)(13 129)(14 128)(15 127)(16 126)(17 125)(18 124)(19 123)(20 122)(21 159)(22 158)(23 157)(24 156)(25 155)(26 154)(27 153)(28 152)(29 151)(30 150)(31 149)(32 148)(33 147)(34 146)(35 145)(36 144)(37 143)(38 142)(39 141)(40 160)(41 161)(42 180)(43 179)(44 178)(45 177)(46 176)(47 175)(48 174)(49 173)(50 172)(51 171)(52 170)(53 169)(54 168)(55 167)(56 166)(57 165)(58 164)(59 163)(60 162)(61 182)(62 181)(63 200)(64 199)(65 198)(66 197)(67 196)(68 195)(69 194)(70 193)(71 192)(72 191)(73 190)(74 189)(75 188)(76 187)(77 186)(78 185)(79 184)(80 183)(81 220)(82 219)(83 218)(84 217)(85 216)(86 215)(87 214)(88 213)(89 212)(90 211)(91 210)(92 209)(93 208)(94 207)(95 206)(96 205)(97 204)(98 203)(99 202)(100 201)(101 229)(102 228)(103 227)(104 226)(105 225)(106 224)(107 223)(108 222)(109 221)(110 240)(111 239)(112 238)(113 237)(114 236)(115 235)(116 234)(117 233)(118 232)(119 231)(120 230)
(1 16 11 6)(2 17 12 7)(3 18 13 8)(4 19 14 9)(5 20 15 10)(21 36 31 26)(22 37 32 27)(23 38 33 28)(24 39 34 29)(25 40 35 30)(41 56 51 46)(42 57 52 47)(43 58 53 48)(44 59 54 49)(45 60 55 50)(61 76 71 66)(62 77 72 67)(63 78 73 68)(64 79 74 69)(65 80 75 70)(81 96 91 86)(82 97 92 87)(83 98 93 88)(84 99 94 89)(85 100 95 90)(101 116 111 106)(102 117 112 107)(103 118 113 108)(104 119 114 109)(105 120 115 110)(121 126 131 136)(122 127 132 137)(123 128 133 138)(124 129 134 139)(125 130 135 140)(141 146 151 156)(142 147 152 157)(143 148 153 158)(144 149 154 159)(145 150 155 160)(161 166 171 176)(162 167 172 177)(163 168 173 178)(164 169 174 179)(165 170 175 180)(181 186 191 196)(182 187 192 197)(183 188 193 198)(184 189 194 199)(185 190 195 200)(201 206 211 216)(202 207 212 217)(203 208 213 218)(204 209 214 219)(205 210 215 220)(221 226 231 236)(222 227 232 237)(223 228 233 238)(224 229 234 239)(225 230 235 240)
(1 77 11 67)(2 68 12 78)(3 79 13 69)(4 70 14 80)(5 61 15 71)(6 72 16 62)(7 63 17 73)(8 74 18 64)(9 65 19 75)(10 76 20 66)(21 97 31 87)(22 88 32 98)(23 99 33 89)(24 90 34 100)(25 81 35 91)(26 92 36 82)(27 83 37 93)(28 94 38 84)(29 85 39 95)(30 96 40 86)(41 109 51 119)(42 120 52 110)(43 111 53 101)(44 102 54 112)(45 113 55 103)(46 104 56 114)(47 115 57 105)(48 106 58 116)(49 117 59 107)(50 108 60 118)(121 181 131 191)(122 192 132 182)(123 183 133 193)(124 194 134 184)(125 185 135 195)(126 196 136 186)(127 187 137 197)(128 198 138 188)(129 189 139 199)(130 200 140 190)(141 201 151 211)(142 212 152 202)(143 203 153 213)(144 214 154 204)(145 205 155 215)(146 216 156 206)(147 207 157 217)(148 218 158 208)(149 209 159 219)(150 220 160 210)(161 236 171 226)(162 227 172 237)(163 238 173 228)(164 229 174 239)(165 240 175 230)(166 231 176 221)(167 222 177 232)(168 233 178 223)(169 224 179 234)(170 235 180 225)
G:=sub<Sym(240)| (1,56,29)(2,57,30)(3,58,31)(4,59,32)(5,60,33)(6,41,34)(7,42,35)(8,43,36)(9,44,37)(10,45,38)(11,46,39)(12,47,40)(13,48,21)(14,49,22)(15,50,23)(16,51,24)(17,52,25)(18,53,26)(19,54,27)(20,55,28)(61,118,89)(62,119,90)(63,120,91)(64,101,92)(65,102,93)(66,103,94)(67,104,95)(68,105,96)(69,106,97)(70,107,98)(71,108,99)(72,109,100)(73,110,81)(74,111,82)(75,112,83)(76,113,84)(77,114,85)(78,115,86)(79,116,87)(80,117,88)(121,166,151)(122,167,152)(123,168,153)(124,169,154)(125,170,155)(126,171,156)(127,172,157)(128,173,158)(129,174,159)(130,175,160)(131,176,141)(132,177,142)(133,178,143)(134,179,144)(135,180,145)(136,161,146)(137,162,147)(138,163,148)(139,164,149)(140,165,150)(181,231,211)(182,232,212)(183,233,213)(184,234,214)(185,235,215)(186,236,216)(187,237,217)(188,238,218)(189,239,219)(190,240,220)(191,221,201)(192,222,202)(193,223,203)(194,224,204)(195,225,205)(196,226,206)(197,227,207)(198,228,208)(199,229,209)(200,230,210), (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), (1,121)(2,140)(3,139)(4,138)(5,137)(6,136)(7,135)(8,134)(9,133)(10,132)(11,131)(12,130)(13,129)(14,128)(15,127)(16,126)(17,125)(18,124)(19,123)(20,122)(21,159)(22,158)(23,157)(24,156)(25,155)(26,154)(27,153)(28,152)(29,151)(30,150)(31,149)(32,148)(33,147)(34,146)(35,145)(36,144)(37,143)(38,142)(39,141)(40,160)(41,161)(42,180)(43,179)(44,178)(45,177)(46,176)(47,175)(48,174)(49,173)(50,172)(51,171)(52,170)(53,169)(54,168)(55,167)(56,166)(57,165)(58,164)(59,163)(60,162)(61,182)(62,181)(63,200)(64,199)(65,198)(66,197)(67,196)(68,195)(69,194)(70,193)(71,192)(72,191)(73,190)(74,189)(75,188)(76,187)(77,186)(78,185)(79,184)(80,183)(81,220)(82,219)(83,218)(84,217)(85,216)(86,215)(87,214)(88,213)(89,212)(90,211)(91,210)(92,209)(93,208)(94,207)(95,206)(96,205)(97,204)(98,203)(99,202)(100,201)(101,229)(102,228)(103,227)(104,226)(105,225)(106,224)(107,223)(108,222)(109,221)(110,240)(111,239)(112,238)(113,237)(114,236)(115,235)(116,234)(117,233)(118,232)(119,231)(120,230), (1,16,11,6)(2,17,12,7)(3,18,13,8)(4,19,14,9)(5,20,15,10)(21,36,31,26)(22,37,32,27)(23,38,33,28)(24,39,34,29)(25,40,35,30)(41,56,51,46)(42,57,52,47)(43,58,53,48)(44,59,54,49)(45,60,55,50)(61,76,71,66)(62,77,72,67)(63,78,73,68)(64,79,74,69)(65,80,75,70)(81,96,91,86)(82,97,92,87)(83,98,93,88)(84,99,94,89)(85,100,95,90)(101,116,111,106)(102,117,112,107)(103,118,113,108)(104,119,114,109)(105,120,115,110)(121,126,131,136)(122,127,132,137)(123,128,133,138)(124,129,134,139)(125,130,135,140)(141,146,151,156)(142,147,152,157)(143,148,153,158)(144,149,154,159)(145,150,155,160)(161,166,171,176)(162,167,172,177)(163,168,173,178)(164,169,174,179)(165,170,175,180)(181,186,191,196)(182,187,192,197)(183,188,193,198)(184,189,194,199)(185,190,195,200)(201,206,211,216)(202,207,212,217)(203,208,213,218)(204,209,214,219)(205,210,215,220)(221,226,231,236)(222,227,232,237)(223,228,233,238)(224,229,234,239)(225,230,235,240), (1,77,11,67)(2,68,12,78)(3,79,13,69)(4,70,14,80)(5,61,15,71)(6,72,16,62)(7,63,17,73)(8,74,18,64)(9,65,19,75)(10,76,20,66)(21,97,31,87)(22,88,32,98)(23,99,33,89)(24,90,34,100)(25,81,35,91)(26,92,36,82)(27,83,37,93)(28,94,38,84)(29,85,39,95)(30,96,40,86)(41,109,51,119)(42,120,52,110)(43,111,53,101)(44,102,54,112)(45,113,55,103)(46,104,56,114)(47,115,57,105)(48,106,58,116)(49,117,59,107)(50,108,60,118)(121,181,131,191)(122,192,132,182)(123,183,133,193)(124,194,134,184)(125,185,135,195)(126,196,136,186)(127,187,137,197)(128,198,138,188)(129,189,139,199)(130,200,140,190)(141,201,151,211)(142,212,152,202)(143,203,153,213)(144,214,154,204)(145,205,155,215)(146,216,156,206)(147,207,157,217)(148,218,158,208)(149,209,159,219)(150,220,160,210)(161,236,171,226)(162,227,172,237)(163,238,173,228)(164,229,174,239)(165,240,175,230)(166,231,176,221)(167,222,177,232)(168,233,178,223)(169,224,179,234)(170,235,180,225)>;
G:=Group( (1,56,29)(2,57,30)(3,58,31)(4,59,32)(5,60,33)(6,41,34)(7,42,35)(8,43,36)(9,44,37)(10,45,38)(11,46,39)(12,47,40)(13,48,21)(14,49,22)(15,50,23)(16,51,24)(17,52,25)(18,53,26)(19,54,27)(20,55,28)(61,118,89)(62,119,90)(63,120,91)(64,101,92)(65,102,93)(66,103,94)(67,104,95)(68,105,96)(69,106,97)(70,107,98)(71,108,99)(72,109,100)(73,110,81)(74,111,82)(75,112,83)(76,113,84)(77,114,85)(78,115,86)(79,116,87)(80,117,88)(121,166,151)(122,167,152)(123,168,153)(124,169,154)(125,170,155)(126,171,156)(127,172,157)(128,173,158)(129,174,159)(130,175,160)(131,176,141)(132,177,142)(133,178,143)(134,179,144)(135,180,145)(136,161,146)(137,162,147)(138,163,148)(139,164,149)(140,165,150)(181,231,211)(182,232,212)(183,233,213)(184,234,214)(185,235,215)(186,236,216)(187,237,217)(188,238,218)(189,239,219)(190,240,220)(191,221,201)(192,222,202)(193,223,203)(194,224,204)(195,225,205)(196,226,206)(197,227,207)(198,228,208)(199,229,209)(200,230,210), (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), (1,121)(2,140)(3,139)(4,138)(5,137)(6,136)(7,135)(8,134)(9,133)(10,132)(11,131)(12,130)(13,129)(14,128)(15,127)(16,126)(17,125)(18,124)(19,123)(20,122)(21,159)(22,158)(23,157)(24,156)(25,155)(26,154)(27,153)(28,152)(29,151)(30,150)(31,149)(32,148)(33,147)(34,146)(35,145)(36,144)(37,143)(38,142)(39,141)(40,160)(41,161)(42,180)(43,179)(44,178)(45,177)(46,176)(47,175)(48,174)(49,173)(50,172)(51,171)(52,170)(53,169)(54,168)(55,167)(56,166)(57,165)(58,164)(59,163)(60,162)(61,182)(62,181)(63,200)(64,199)(65,198)(66,197)(67,196)(68,195)(69,194)(70,193)(71,192)(72,191)(73,190)(74,189)(75,188)(76,187)(77,186)(78,185)(79,184)(80,183)(81,220)(82,219)(83,218)(84,217)(85,216)(86,215)(87,214)(88,213)(89,212)(90,211)(91,210)(92,209)(93,208)(94,207)(95,206)(96,205)(97,204)(98,203)(99,202)(100,201)(101,229)(102,228)(103,227)(104,226)(105,225)(106,224)(107,223)(108,222)(109,221)(110,240)(111,239)(112,238)(113,237)(114,236)(115,235)(116,234)(117,233)(118,232)(119,231)(120,230), (1,16,11,6)(2,17,12,7)(3,18,13,8)(4,19,14,9)(5,20,15,10)(21,36,31,26)(22,37,32,27)(23,38,33,28)(24,39,34,29)(25,40,35,30)(41,56,51,46)(42,57,52,47)(43,58,53,48)(44,59,54,49)(45,60,55,50)(61,76,71,66)(62,77,72,67)(63,78,73,68)(64,79,74,69)(65,80,75,70)(81,96,91,86)(82,97,92,87)(83,98,93,88)(84,99,94,89)(85,100,95,90)(101,116,111,106)(102,117,112,107)(103,118,113,108)(104,119,114,109)(105,120,115,110)(121,126,131,136)(122,127,132,137)(123,128,133,138)(124,129,134,139)(125,130,135,140)(141,146,151,156)(142,147,152,157)(143,148,153,158)(144,149,154,159)(145,150,155,160)(161,166,171,176)(162,167,172,177)(163,168,173,178)(164,169,174,179)(165,170,175,180)(181,186,191,196)(182,187,192,197)(183,188,193,198)(184,189,194,199)(185,190,195,200)(201,206,211,216)(202,207,212,217)(203,208,213,218)(204,209,214,219)(205,210,215,220)(221,226,231,236)(222,227,232,237)(223,228,233,238)(224,229,234,239)(225,230,235,240), (1,77,11,67)(2,68,12,78)(3,79,13,69)(4,70,14,80)(5,61,15,71)(6,72,16,62)(7,63,17,73)(8,74,18,64)(9,65,19,75)(10,76,20,66)(21,97,31,87)(22,88,32,98)(23,99,33,89)(24,90,34,100)(25,81,35,91)(26,92,36,82)(27,83,37,93)(28,94,38,84)(29,85,39,95)(30,96,40,86)(41,109,51,119)(42,120,52,110)(43,111,53,101)(44,102,54,112)(45,113,55,103)(46,104,56,114)(47,115,57,105)(48,106,58,116)(49,117,59,107)(50,108,60,118)(121,181,131,191)(122,192,132,182)(123,183,133,193)(124,194,134,184)(125,185,135,195)(126,196,136,186)(127,187,137,197)(128,198,138,188)(129,189,139,199)(130,200,140,190)(141,201,151,211)(142,212,152,202)(143,203,153,213)(144,214,154,204)(145,205,155,215)(146,216,156,206)(147,207,157,217)(148,218,158,208)(149,209,159,219)(150,220,160,210)(161,236,171,226)(162,227,172,237)(163,238,173,228)(164,229,174,239)(165,240,175,230)(166,231,176,221)(167,222,177,232)(168,233,178,223)(169,224,179,234)(170,235,180,225) );
G=PermutationGroup([[(1,56,29),(2,57,30),(3,58,31),(4,59,32),(5,60,33),(6,41,34),(7,42,35),(8,43,36),(9,44,37),(10,45,38),(11,46,39),(12,47,40),(13,48,21),(14,49,22),(15,50,23),(16,51,24),(17,52,25),(18,53,26),(19,54,27),(20,55,28),(61,118,89),(62,119,90),(63,120,91),(64,101,92),(65,102,93),(66,103,94),(67,104,95),(68,105,96),(69,106,97),(70,107,98),(71,108,99),(72,109,100),(73,110,81),(74,111,82),(75,112,83),(76,113,84),(77,114,85),(78,115,86),(79,116,87),(80,117,88),(121,166,151),(122,167,152),(123,168,153),(124,169,154),(125,170,155),(126,171,156),(127,172,157),(128,173,158),(129,174,159),(130,175,160),(131,176,141),(132,177,142),(133,178,143),(134,179,144),(135,180,145),(136,161,146),(137,162,147),(138,163,148),(139,164,149),(140,165,150),(181,231,211),(182,232,212),(183,233,213),(184,234,214),(185,235,215),(186,236,216),(187,237,217),(188,238,218),(189,239,219),(190,240,220),(191,221,201),(192,222,202),(193,223,203),(194,224,204),(195,225,205),(196,226,206),(197,227,207),(198,228,208),(199,229,209),(200,230,210)], [(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)], [(1,121),(2,140),(3,139),(4,138),(5,137),(6,136),(7,135),(8,134),(9,133),(10,132),(11,131),(12,130),(13,129),(14,128),(15,127),(16,126),(17,125),(18,124),(19,123),(20,122),(21,159),(22,158),(23,157),(24,156),(25,155),(26,154),(27,153),(28,152),(29,151),(30,150),(31,149),(32,148),(33,147),(34,146),(35,145),(36,144),(37,143),(38,142),(39,141),(40,160),(41,161),(42,180),(43,179),(44,178),(45,177),(46,176),(47,175),(48,174),(49,173),(50,172),(51,171),(52,170),(53,169),(54,168),(55,167),(56,166),(57,165),(58,164),(59,163),(60,162),(61,182),(62,181),(63,200),(64,199),(65,198),(66,197),(67,196),(68,195),(69,194),(70,193),(71,192),(72,191),(73,190),(74,189),(75,188),(76,187),(77,186),(78,185),(79,184),(80,183),(81,220),(82,219),(83,218),(84,217),(85,216),(86,215),(87,214),(88,213),(89,212),(90,211),(91,210),(92,209),(93,208),(94,207),(95,206),(96,205),(97,204),(98,203),(99,202),(100,201),(101,229),(102,228),(103,227),(104,226),(105,225),(106,224),(107,223),(108,222),(109,221),(110,240),(111,239),(112,238),(113,237),(114,236),(115,235),(116,234),(117,233),(118,232),(119,231),(120,230)], [(1,16,11,6),(2,17,12,7),(3,18,13,8),(4,19,14,9),(5,20,15,10),(21,36,31,26),(22,37,32,27),(23,38,33,28),(24,39,34,29),(25,40,35,30),(41,56,51,46),(42,57,52,47),(43,58,53,48),(44,59,54,49),(45,60,55,50),(61,76,71,66),(62,77,72,67),(63,78,73,68),(64,79,74,69),(65,80,75,70),(81,96,91,86),(82,97,92,87),(83,98,93,88),(84,99,94,89),(85,100,95,90),(101,116,111,106),(102,117,112,107),(103,118,113,108),(104,119,114,109),(105,120,115,110),(121,126,131,136),(122,127,132,137),(123,128,133,138),(124,129,134,139),(125,130,135,140),(141,146,151,156),(142,147,152,157),(143,148,153,158),(144,149,154,159),(145,150,155,160),(161,166,171,176),(162,167,172,177),(163,168,173,178),(164,169,174,179),(165,170,175,180),(181,186,191,196),(182,187,192,197),(183,188,193,198),(184,189,194,199),(185,190,195,200),(201,206,211,216),(202,207,212,217),(203,208,213,218),(204,209,214,219),(205,210,215,220),(221,226,231,236),(222,227,232,237),(223,228,233,238),(224,229,234,239),(225,230,235,240)], [(1,77,11,67),(2,68,12,78),(3,79,13,69),(4,70,14,80),(5,61,15,71),(6,72,16,62),(7,63,17,73),(8,74,18,64),(9,65,19,75),(10,76,20,66),(21,97,31,87),(22,88,32,98),(23,99,33,89),(24,90,34,100),(25,81,35,91),(26,92,36,82),(27,83,37,93),(28,94,38,84),(29,85,39,95),(30,96,40,86),(41,109,51,119),(42,120,52,110),(43,111,53,101),(44,102,54,112),(45,113,55,103),(46,104,56,114),(47,115,57,105),(48,106,58,116),(49,117,59,107),(50,108,60,118),(121,181,131,191),(122,192,132,182),(123,183,133,193),(124,194,134,184),(125,185,135,195),(126,196,136,186),(127,187,137,197),(128,198,138,188),(129,189,139,199),(130,200,140,190),(141,201,151,211),(142,212,152,202),(143,203,153,213),(144,214,154,204),(145,205,155,215),(146,216,156,206),(147,207,157,217),(148,218,158,208),(149,209,159,219),(150,220,160,210),(161,236,171,226),(162,227,172,237),(163,238,173,228),(164,229,174,239),(165,240,175,230),(166,231,176,221),(167,222,177,232),(168,233,178,223),(169,224,179,234),(170,235,180,225)]])
93 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 15A | 15B | 15C | 15D | 20A | ··· | 20L | 24A | 24B | 24C | 24D | 30A | ··· | 30L | 60A | ··· | 60X |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 24 | 24 | 24 | 24 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 20 | 1 | 1 | 2 | 2 | 4 | 4 | 20 | 2 | 2 | 1 | 1 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 |
93 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 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | ||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | D4 | D4 | D5 | D10 | D10 | C3×D4 | C3×D4 | C3×D5 | C5⋊D4 | C5⋊D4 | C6×D5 | C6×D5 | C3×C5⋊D4 | C3×C5⋊D4 | C8.C22 | C3×C8.C22 | C20.C23 | C3×C20.C23 |
| kernel | C3×C20.C23 | C3×C4.Dic5 | C3×Q8⋊D5 | C3×C5⋊Q16 | C3×C4○D20 | Q8×C30 | C20.C23 | C4.Dic5 | Q8⋊D5 | C5⋊Q16 | C4○D20 | Q8×C10 | C60 | C2×C30 | C6×Q8 | C2×C12 | C3×Q8 | C20 | C2×C10 | C2×Q8 | C12 | C2×C6 | C2×C4 | Q8 | C4 | C22 | C15 | C5 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 1 | 2 | 4 | 8 |
Matrix representation of C3×C20.C23 ►in GL6(𝔽241)
| 225 | 0 | 0 | 0 | 0 | 0 |
| 0 | 225 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 205 | 239 | 0 | 0 |
| 0 | 0 | 91 | 36 | 0 | 0 |
| 0 | 0 | 84 | 181 | 0 | 154 |
| 0 | 0 | 166 | 181 | 87 | 0 |
| 68 | 238 | 0 | 0 | 0 | 0 |
| 95 | 173 | 0 | 0 | 0 | 0 |
| 0 | 0 | 159 | 0 | 67 | 0 |
| 0 | 0 | 0 | 0 | 240 | 1 |
| 0 | 0 | 209 | 0 | 82 | 0 |
| 0 | 0 | 209 | 1 | 82 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 174 | 0 | 0 |
| 0 | 0 | 36 | 1 | 0 | 0 |
| 0 | 0 | 0 | 159 | 0 | 240 |
| 0 | 0 | 181 | 159 | 1 | 0 |
| 210 | 74 | 0 | 0 | 0 | 0 |
| 215 | 31 | 0 | 0 | 0 | 0 |
| 0 | 0 | 177 | 0 | 0 | 0 |
| 0 | 0 | 135 | 64 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 64 |
| 0 | 0 | 16 | 0 | 64 | 0 |
G:=sub<GL(6,GF(241))| [225,0,0,0,0,0,0,225,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,205,91,84,166,0,0,239,36,181,181,0,0,0,0,0,87,0,0,0,0,154,0],[68,95,0,0,0,0,238,173,0,0,0,0,0,0,159,0,209,209,0,0,0,0,0,1,0,0,67,240,82,82,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,36,0,181,0,0,174,1,159,159,0,0,0,0,0,1,0,0,0,0,240,0],[210,215,0,0,0,0,74,31,0,0,0,0,0,0,177,135,16,16,0,0,0,64,0,0,0,0,0,0,0,64,0,0,0,0,64,0] >;
C3×C20.C23 in GAP, Magma, Sage, TeX
C_3\times C_{20}.C_2^3 % in TeX
G:=Group("C3xC20.C2^3"); // GroupNames label
G:=SmallGroup(480,735);
// by ID
G=gap.SmallGroup(480,735);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,344,590,555,268,2524,648,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^20=c^2=1,d^2=e^2=b^10,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e^-1=b^11,c*d=d*c,e*c*e^-1=b^5*c,e*d*e^-1=b^10*d>;
// generators/relations