direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C23.9D6, D6⋊C4⋊5C10, D6.4(C5×D4), C4⋊Dic3⋊4C10, C6.19(D4×C10), (S3×C10).40D4, (C2×C20).272D6, C10.172(S3×D4), C30.355(C2×D4), C23.9(S3×C10), Dic3⋊C4⋊10C10, C6.D4⋊4C10, (C22×C10).21D6, C30.245(C4○D4), (C2×C60).327C22, (C2×C30).403C23, C10.115(C4○D12), C15⋊29(C22.D4), C10.110(D4⋊2S3), (C22×C30).118C22, (C10×Dic3).139C22, C2.8(C5×S3×D4), (S3×C2×C4)⋊10C10, (S3×C2×C20)⋊26C2, C6.7(C5×C4○D4), (C5×D6⋊C4)⋊21C2, C22⋊C4⋊3(C5×S3), (C2×C4).6(S3×C10), (C5×C22⋊C4)⋊11S3, (C3×C22⋊C4)⋊5C10, (C2×C12).2(C2×C10), (C5×C4⋊Dic3)⋊22C2, C2.10(C5×C4○D12), C2.8(C5×D4⋊2S3), (C2×C3⋊D4).3C10, C22.42(S3×C2×C10), (C15×C22⋊C4)⋊19C2, C3⋊1(C5×C22.D4), (C5×Dic3⋊C4)⋊32C2, (C10×C3⋊D4).10C2, (C5×C6.D4)⋊20C2, (S3×C2×C10).108C22, (C2×C6).24(C22×C10), (C22×C6).13(C2×C10), (C22×S3).18(C2×C10), (C2×C10).337(C22×S3), (C2×Dic3).21(C2×C10), SmallGroup(480,762)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C23.9D6
G = < a,b,c,d,e,f | a5=b2=c2=d2=1, e6=f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=bd=db, fbf-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >
Subgroups: 372 in 156 conjugacy classes, 62 normal (58 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, C10, C10, Dic3, C12, D6, D6, C2×C6, C2×C6, C15, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C20, C2×C10, C2×C10, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C5×S3, C30, C30, C22.D4, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, S3×C2×C4, C2×C3⋊D4, C5×Dic3, C60, S3×C10, S3×C10, C2×C30, C2×C30, C5×C22⋊C4, C5×C22⋊C4, C5×C4⋊C4, C22×C20, D4×C10, C23.9D6, S3×C20, C10×Dic3, C5×C3⋊D4, C2×C60, S3×C2×C10, C22×C30, C5×C22.D4, C5×Dic3⋊C4, C5×C4⋊Dic3, C5×D6⋊C4, C5×C6.D4, C15×C22⋊C4, S3×C2×C20, C10×C3⋊D4, C5×C23.9D6
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, C2×D4, C4○D4, C2×C10, C22×S3, C5×S3, C22.D4, C5×D4, C22×C10, C4○D12, S3×D4, D4⋊2S3, S3×C10, D4×C10, C5×C4○D4, C23.9D6, S3×C2×C10, C5×C22.D4, C5×C4○D12, C5×S3×D4, C5×D4⋊2S3, C5×C23.9D6
►On 240 points
Generators in S240
(1 174 224 202 187)(2 175 225 203 188)(3 176 226 204 189)(4 177 227 193 190)(5 178 228 194 191)(6 179 217 195 192)(7 180 218 196 181)(8 169 219 197 182)(9 170 220 198 183)(10 171 221 199 184)(11 172 222 200 185)(12 173 223 201 186)(13 83 144 65 96)(14 84 133 66 85)(15 73 134 67 86)(16 74 135 68 87)(17 75 136 69 88)(18 76 137 70 89)(19 77 138 71 90)(20 78 139 72 91)(21 79 140 61 92)(22 80 141 62 93)(23 81 142 63 94)(24 82 143 64 95)(25 154 236 41 57)(26 155 237 42 58)(27 156 238 43 59)(28 145 239 44 60)(29 146 240 45 49)(30 147 229 46 50)(31 148 230 47 51)(32 149 231 48 52)(33 150 232 37 53)(34 151 233 38 54)(35 152 234 39 55)(36 153 235 40 56)(97 216 167 122 113)(98 205 168 123 114)(99 206 157 124 115)(100 207 158 125 116)(101 208 159 126 117)(102 209 160 127 118)(103 210 161 128 119)(104 211 162 129 120)(105 212 163 130 109)(106 213 164 131 110)(107 214 165 132 111)(108 215 166 121 112)
(1 149)(2 76)(3 151)(4 78)(5 153)(6 80)(7 155)(8 82)(9 145)(10 84)(11 147)(12 74)(13 120)(14 184)(15 110)(16 186)(17 112)(18 188)(19 114)(20 190)(21 116)(22 192)(23 118)(24 182)(25 117)(26 181)(27 119)(28 183)(29 109)(30 185)(31 111)(32 187)(33 113)(34 189)(35 115)(36 191)(37 167)(38 226)(39 157)(40 228)(41 159)(42 218)(43 161)(44 220)(45 163)(46 222)(47 165)(48 224)(49 130)(50 200)(51 132)(52 202)(53 122)(54 204)(55 124)(56 194)(57 126)(58 196)(59 128)(60 198)(61 158)(62 217)(63 160)(64 219)(65 162)(66 221)(67 164)(68 223)(69 166)(70 225)(71 168)(72 227)(73 106)(75 108)(77 98)(79 100)(81 102)(83 104)(85 199)(86 131)(87 201)(88 121)(89 203)(90 123)(91 193)(92 125)(93 195)(94 127)(95 197)(96 129)(97 150)(99 152)(101 154)(103 156)(105 146)(107 148)(133 171)(134 213)(135 173)(136 215)(137 175)(138 205)(139 177)(140 207)(141 179)(142 209)(143 169)(144 211)(170 239)(172 229)(174 231)(176 233)(178 235)(180 237)(206 234)(208 236)(210 238)(212 240)(214 230)(216 232)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)(97 103)(98 104)(99 105)(100 106)(101 107)(102 108)(109 115)(110 116)(111 117)(112 118)(113 119)(114 120)(121 127)(122 128)(123 129)(124 130)(125 131)(126 132)(133 139)(134 140)(135 141)(136 142)(137 143)(138 144)(145 151)(146 152)(147 153)(148 154)(149 155)(150 156)(157 163)(158 164)(159 165)(160 166)(161 167)(162 168)(169 175)(170 176)(171 177)(172 178)(173 179)(174 180)(181 187)(182 188)(183 189)(184 190)(185 191)(186 192)(193 199)(194 200)(195 201)(196 202)(197 203)(198 204)(205 211)(206 212)(207 213)(208 214)(209 215)(210 216)(217 223)(218 224)(219 225)(220 226)(221 227)(222 228)(229 235)(230 236)(231 237)(232 238)(233 239)(234 240)
(1 108)(2 97)(3 98)(4 99)(5 100)(6 101)(7 102)(8 103)(9 104)(10 105)(11 106)(12 107)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(19 34)(20 35)(21 36)(22 25)(23 26)(24 27)(37 70)(38 71)(39 72)(40 61)(41 62)(42 63)(43 64)(44 65)(45 66)(46 67)(47 68)(48 69)(49 85)(50 86)(51 87)(52 88)(53 89)(54 90)(55 91)(56 92)(57 93)(58 94)(59 95)(60 96)(73 147)(74 148)(75 149)(76 150)(77 151)(78 152)(79 153)(80 154)(81 155)(82 156)(83 145)(84 146)(109 184)(110 185)(111 186)(112 187)(113 188)(114 189)(115 190)(116 191)(117 192)(118 181)(119 182)(120 183)(121 202)(122 203)(123 204)(124 193)(125 194)(126 195)(127 196)(128 197)(129 198)(130 199)(131 200)(132 201)(133 240)(134 229)(135 230)(136 231)(137 232)(138 233)(139 234)(140 235)(141 236)(142 237)(143 238)(144 239)(157 227)(158 228)(159 217)(160 218)(161 219)(162 220)(163 221)(164 222)(165 223)(166 224)(167 225)(168 226)(169 210)(170 211)(171 212)(172 213)(173 214)(174 215)(175 216)(176 205)(177 206)(178 207)(179 208)(180 209)
(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 6 7 12)(2 11 8 5)(3 4 9 10)(13 20 19 14)(15 18 21 24)(16 23 22 17)(25 32 31 26)(27 30 33 36)(28 35 34 29)(37 40 43 46)(38 45 44 39)(41 48 47 42)(49 60 55 54)(50 53 56 59)(51 58 57 52)(61 64 67 70)(62 69 68 63)(65 72 71 66)(73 76 79 82)(74 81 80 75)(77 84 83 78)(85 96 91 90)(86 89 92 95)(87 94 93 88)(97 106 103 100)(98 99 104 105)(101 102 107 108)(109 114 115 120)(110 119 116 113)(111 112 117 118)(121 126 127 132)(122 131 128 125)(123 124 129 130)(133 144 139 138)(134 137 140 143)(135 142 141 136)(145 152 151 146)(147 150 153 156)(148 155 154 149)(157 162 163 168)(158 167 164 161)(159 160 165 166)(169 178 175 172)(170 171 176 177)(173 174 179 180)(181 186 187 192)(182 191 188 185)(183 184 189 190)(193 198 199 204)(194 203 200 197)(195 196 201 202)(205 206 211 212)(207 216 213 210)(208 209 214 215)(217 218 223 224)(219 228 225 222)(220 221 226 227)(229 232 235 238)(230 237 236 231)(233 240 239 234)
G:=sub<Sym(240)| (1,174,224,202,187)(2,175,225,203,188)(3,176,226,204,189)(4,177,227,193,190)(5,178,228,194,191)(6,179,217,195,192)(7,180,218,196,181)(8,169,219,197,182)(9,170,220,198,183)(10,171,221,199,184)(11,172,222,200,185)(12,173,223,201,186)(13,83,144,65,96)(14,84,133,66,85)(15,73,134,67,86)(16,74,135,68,87)(17,75,136,69,88)(18,76,137,70,89)(19,77,138,71,90)(20,78,139,72,91)(21,79,140,61,92)(22,80,141,62,93)(23,81,142,63,94)(24,82,143,64,95)(25,154,236,41,57)(26,155,237,42,58)(27,156,238,43,59)(28,145,239,44,60)(29,146,240,45,49)(30,147,229,46,50)(31,148,230,47,51)(32,149,231,48,52)(33,150,232,37,53)(34,151,233,38,54)(35,152,234,39,55)(36,153,235,40,56)(97,216,167,122,113)(98,205,168,123,114)(99,206,157,124,115)(100,207,158,125,116)(101,208,159,126,117)(102,209,160,127,118)(103,210,161,128,119)(104,211,162,129,120)(105,212,163,130,109)(106,213,164,131,110)(107,214,165,132,111)(108,215,166,121,112), (1,149)(2,76)(3,151)(4,78)(5,153)(6,80)(7,155)(8,82)(9,145)(10,84)(11,147)(12,74)(13,120)(14,184)(15,110)(16,186)(17,112)(18,188)(19,114)(20,190)(21,116)(22,192)(23,118)(24,182)(25,117)(26,181)(27,119)(28,183)(29,109)(30,185)(31,111)(32,187)(33,113)(34,189)(35,115)(36,191)(37,167)(38,226)(39,157)(40,228)(41,159)(42,218)(43,161)(44,220)(45,163)(46,222)(47,165)(48,224)(49,130)(50,200)(51,132)(52,202)(53,122)(54,204)(55,124)(56,194)(57,126)(58,196)(59,128)(60,198)(61,158)(62,217)(63,160)(64,219)(65,162)(66,221)(67,164)(68,223)(69,166)(70,225)(71,168)(72,227)(73,106)(75,108)(77,98)(79,100)(81,102)(83,104)(85,199)(86,131)(87,201)(88,121)(89,203)(90,123)(91,193)(92,125)(93,195)(94,127)(95,197)(96,129)(97,150)(99,152)(101,154)(103,156)(105,146)(107,148)(133,171)(134,213)(135,173)(136,215)(137,175)(138,205)(139,177)(140,207)(141,179)(142,209)(143,169)(144,211)(170,239)(172,229)(174,231)(176,233)(178,235)(180,237)(206,234)(208,236)(210,238)(212,240)(214,230)(216,232), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96)(97,103)(98,104)(99,105)(100,106)(101,107)(102,108)(109,115)(110,116)(111,117)(112,118)(113,119)(114,120)(121,127)(122,128)(123,129)(124,130)(125,131)(126,132)(133,139)(134,140)(135,141)(136,142)(137,143)(138,144)(145,151)(146,152)(147,153)(148,154)(149,155)(150,156)(157,163)(158,164)(159,165)(160,166)(161,167)(162,168)(169,175)(170,176)(171,177)(172,178)(173,179)(174,180)(181,187)(182,188)(183,189)(184,190)(185,191)(186,192)(193,199)(194,200)(195,201)(196,202)(197,203)(198,204)(205,211)(206,212)(207,213)(208,214)(209,215)(210,216)(217,223)(218,224)(219,225)(220,226)(221,227)(222,228)(229,235)(230,236)(231,237)(232,238)(233,239)(234,240), (1,108)(2,97)(3,98)(4,99)(5,100)(6,101)(7,102)(8,103)(9,104)(10,105)(11,106)(12,107)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35)(21,36)(22,25)(23,26)(24,27)(37,70)(38,71)(39,72)(40,61)(41,62)(42,63)(43,64)(44,65)(45,66)(46,67)(47,68)(48,69)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(55,91)(56,92)(57,93)(58,94)(59,95)(60,96)(73,147)(74,148)(75,149)(76,150)(77,151)(78,152)(79,153)(80,154)(81,155)(82,156)(83,145)(84,146)(109,184)(110,185)(111,186)(112,187)(113,188)(114,189)(115,190)(116,191)(117,192)(118,181)(119,182)(120,183)(121,202)(122,203)(123,204)(124,193)(125,194)(126,195)(127,196)(128,197)(129,198)(130,199)(131,200)(132,201)(133,240)(134,229)(135,230)(136,231)(137,232)(138,233)(139,234)(140,235)(141,236)(142,237)(143,238)(144,239)(157,227)(158,228)(159,217)(160,218)(161,219)(162,220)(163,221)(164,222)(165,223)(166,224)(167,225)(168,226)(169,210)(170,211)(171,212)(172,213)(173,214)(174,215)(175,216)(176,205)(177,206)(178,207)(179,208)(180,209), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,20,19,14)(15,18,21,24)(16,23,22,17)(25,32,31,26)(27,30,33,36)(28,35,34,29)(37,40,43,46)(38,45,44,39)(41,48,47,42)(49,60,55,54)(50,53,56,59)(51,58,57,52)(61,64,67,70)(62,69,68,63)(65,72,71,66)(73,76,79,82)(74,81,80,75)(77,84,83,78)(85,96,91,90)(86,89,92,95)(87,94,93,88)(97,106,103,100)(98,99,104,105)(101,102,107,108)(109,114,115,120)(110,119,116,113)(111,112,117,118)(121,126,127,132)(122,131,128,125)(123,124,129,130)(133,144,139,138)(134,137,140,143)(135,142,141,136)(145,152,151,146)(147,150,153,156)(148,155,154,149)(157,162,163,168)(158,167,164,161)(159,160,165,166)(169,178,175,172)(170,171,176,177)(173,174,179,180)(181,186,187,192)(182,191,188,185)(183,184,189,190)(193,198,199,204)(194,203,200,197)(195,196,201,202)(205,206,211,212)(207,216,213,210)(208,209,214,215)(217,218,223,224)(219,228,225,222)(220,221,226,227)(229,232,235,238)(230,237,236,231)(233,240,239,234)>;
G:=Group( (1,174,224,202,187)(2,175,225,203,188)(3,176,226,204,189)(4,177,227,193,190)(5,178,228,194,191)(6,179,217,195,192)(7,180,218,196,181)(8,169,219,197,182)(9,170,220,198,183)(10,171,221,199,184)(11,172,222,200,185)(12,173,223,201,186)(13,83,144,65,96)(14,84,133,66,85)(15,73,134,67,86)(16,74,135,68,87)(17,75,136,69,88)(18,76,137,70,89)(19,77,138,71,90)(20,78,139,72,91)(21,79,140,61,92)(22,80,141,62,93)(23,81,142,63,94)(24,82,143,64,95)(25,154,236,41,57)(26,155,237,42,58)(27,156,238,43,59)(28,145,239,44,60)(29,146,240,45,49)(30,147,229,46,50)(31,148,230,47,51)(32,149,231,48,52)(33,150,232,37,53)(34,151,233,38,54)(35,152,234,39,55)(36,153,235,40,56)(97,216,167,122,113)(98,205,168,123,114)(99,206,157,124,115)(100,207,158,125,116)(101,208,159,126,117)(102,209,160,127,118)(103,210,161,128,119)(104,211,162,129,120)(105,212,163,130,109)(106,213,164,131,110)(107,214,165,132,111)(108,215,166,121,112), (1,149)(2,76)(3,151)(4,78)(5,153)(6,80)(7,155)(8,82)(9,145)(10,84)(11,147)(12,74)(13,120)(14,184)(15,110)(16,186)(17,112)(18,188)(19,114)(20,190)(21,116)(22,192)(23,118)(24,182)(25,117)(26,181)(27,119)(28,183)(29,109)(30,185)(31,111)(32,187)(33,113)(34,189)(35,115)(36,191)(37,167)(38,226)(39,157)(40,228)(41,159)(42,218)(43,161)(44,220)(45,163)(46,222)(47,165)(48,224)(49,130)(50,200)(51,132)(52,202)(53,122)(54,204)(55,124)(56,194)(57,126)(58,196)(59,128)(60,198)(61,158)(62,217)(63,160)(64,219)(65,162)(66,221)(67,164)(68,223)(69,166)(70,225)(71,168)(72,227)(73,106)(75,108)(77,98)(79,100)(81,102)(83,104)(85,199)(86,131)(87,201)(88,121)(89,203)(90,123)(91,193)(92,125)(93,195)(94,127)(95,197)(96,129)(97,150)(99,152)(101,154)(103,156)(105,146)(107,148)(133,171)(134,213)(135,173)(136,215)(137,175)(138,205)(139,177)(140,207)(141,179)(142,209)(143,169)(144,211)(170,239)(172,229)(174,231)(176,233)(178,235)(180,237)(206,234)(208,236)(210,238)(212,240)(214,230)(216,232), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96)(97,103)(98,104)(99,105)(100,106)(101,107)(102,108)(109,115)(110,116)(111,117)(112,118)(113,119)(114,120)(121,127)(122,128)(123,129)(124,130)(125,131)(126,132)(133,139)(134,140)(135,141)(136,142)(137,143)(138,144)(145,151)(146,152)(147,153)(148,154)(149,155)(150,156)(157,163)(158,164)(159,165)(160,166)(161,167)(162,168)(169,175)(170,176)(171,177)(172,178)(173,179)(174,180)(181,187)(182,188)(183,189)(184,190)(185,191)(186,192)(193,199)(194,200)(195,201)(196,202)(197,203)(198,204)(205,211)(206,212)(207,213)(208,214)(209,215)(210,216)(217,223)(218,224)(219,225)(220,226)(221,227)(222,228)(229,235)(230,236)(231,237)(232,238)(233,239)(234,240), (1,108)(2,97)(3,98)(4,99)(5,100)(6,101)(7,102)(8,103)(9,104)(10,105)(11,106)(12,107)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35)(21,36)(22,25)(23,26)(24,27)(37,70)(38,71)(39,72)(40,61)(41,62)(42,63)(43,64)(44,65)(45,66)(46,67)(47,68)(48,69)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(55,91)(56,92)(57,93)(58,94)(59,95)(60,96)(73,147)(74,148)(75,149)(76,150)(77,151)(78,152)(79,153)(80,154)(81,155)(82,156)(83,145)(84,146)(109,184)(110,185)(111,186)(112,187)(113,188)(114,189)(115,190)(116,191)(117,192)(118,181)(119,182)(120,183)(121,202)(122,203)(123,204)(124,193)(125,194)(126,195)(127,196)(128,197)(129,198)(130,199)(131,200)(132,201)(133,240)(134,229)(135,230)(136,231)(137,232)(138,233)(139,234)(140,235)(141,236)(142,237)(143,238)(144,239)(157,227)(158,228)(159,217)(160,218)(161,219)(162,220)(163,221)(164,222)(165,223)(166,224)(167,225)(168,226)(169,210)(170,211)(171,212)(172,213)(173,214)(174,215)(175,216)(176,205)(177,206)(178,207)(179,208)(180,209), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,20,19,14)(15,18,21,24)(16,23,22,17)(25,32,31,26)(27,30,33,36)(28,35,34,29)(37,40,43,46)(38,45,44,39)(41,48,47,42)(49,60,55,54)(50,53,56,59)(51,58,57,52)(61,64,67,70)(62,69,68,63)(65,72,71,66)(73,76,79,82)(74,81,80,75)(77,84,83,78)(85,96,91,90)(86,89,92,95)(87,94,93,88)(97,106,103,100)(98,99,104,105)(101,102,107,108)(109,114,115,120)(110,119,116,113)(111,112,117,118)(121,126,127,132)(122,131,128,125)(123,124,129,130)(133,144,139,138)(134,137,140,143)(135,142,141,136)(145,152,151,146)(147,150,153,156)(148,155,154,149)(157,162,163,168)(158,167,164,161)(159,160,165,166)(169,178,175,172)(170,171,176,177)(173,174,179,180)(181,186,187,192)(182,191,188,185)(183,184,189,190)(193,198,199,204)(194,203,200,197)(195,196,201,202)(205,206,211,212)(207,216,213,210)(208,209,214,215)(217,218,223,224)(219,228,225,222)(220,221,226,227)(229,232,235,238)(230,237,236,231)(233,240,239,234) );
G=PermutationGroup([[(1,174,224,202,187),(2,175,225,203,188),(3,176,226,204,189),(4,177,227,193,190),(5,178,228,194,191),(6,179,217,195,192),(7,180,218,196,181),(8,169,219,197,182),(9,170,220,198,183),(10,171,221,199,184),(11,172,222,200,185),(12,173,223,201,186),(13,83,144,65,96),(14,84,133,66,85),(15,73,134,67,86),(16,74,135,68,87),(17,75,136,69,88),(18,76,137,70,89),(19,77,138,71,90),(20,78,139,72,91),(21,79,140,61,92),(22,80,141,62,93),(23,81,142,63,94),(24,82,143,64,95),(25,154,236,41,57),(26,155,237,42,58),(27,156,238,43,59),(28,145,239,44,60),(29,146,240,45,49),(30,147,229,46,50),(31,148,230,47,51),(32,149,231,48,52),(33,150,232,37,53),(34,151,233,38,54),(35,152,234,39,55),(36,153,235,40,56),(97,216,167,122,113),(98,205,168,123,114),(99,206,157,124,115),(100,207,158,125,116),(101,208,159,126,117),(102,209,160,127,118),(103,210,161,128,119),(104,211,162,129,120),(105,212,163,130,109),(106,213,164,131,110),(107,214,165,132,111),(108,215,166,121,112)], [(1,149),(2,76),(3,151),(4,78),(5,153),(6,80),(7,155),(8,82),(9,145),(10,84),(11,147),(12,74),(13,120),(14,184),(15,110),(16,186),(17,112),(18,188),(19,114),(20,190),(21,116),(22,192),(23,118),(24,182),(25,117),(26,181),(27,119),(28,183),(29,109),(30,185),(31,111),(32,187),(33,113),(34,189),(35,115),(36,191),(37,167),(38,226),(39,157),(40,228),(41,159),(42,218),(43,161),(44,220),(45,163),(46,222),(47,165),(48,224),(49,130),(50,200),(51,132),(52,202),(53,122),(54,204),(55,124),(56,194),(57,126),(58,196),(59,128),(60,198),(61,158),(62,217),(63,160),(64,219),(65,162),(66,221),(67,164),(68,223),(69,166),(70,225),(71,168),(72,227),(73,106),(75,108),(77,98),(79,100),(81,102),(83,104),(85,199),(86,131),(87,201),(88,121),(89,203),(90,123),(91,193),(92,125),(93,195),(94,127),(95,197),(96,129),(97,150),(99,152),(101,154),(103,156),(105,146),(107,148),(133,171),(134,213),(135,173),(136,215),(137,175),(138,205),(139,177),(140,207),(141,179),(142,209),(143,169),(144,211),(170,239),(172,229),(174,231),(176,233),(178,235),(180,237),(206,234),(208,236),(210,238),(212,240),(214,230),(216,232)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48),(49,55),(50,56),(51,57),(52,58),(53,59),(54,60),(61,67),(62,68),(63,69),(64,70),(65,71),(66,72),(73,79),(74,80),(75,81),(76,82),(77,83),(78,84),(85,91),(86,92),(87,93),(88,94),(89,95),(90,96),(97,103),(98,104),(99,105),(100,106),(101,107),(102,108),(109,115),(110,116),(111,117),(112,118),(113,119),(114,120),(121,127),(122,128),(123,129),(124,130),(125,131),(126,132),(133,139),(134,140),(135,141),(136,142),(137,143),(138,144),(145,151),(146,152),(147,153),(148,154),(149,155),(150,156),(157,163),(158,164),(159,165),(160,166),(161,167),(162,168),(169,175),(170,176),(171,177),(172,178),(173,179),(174,180),(181,187),(182,188),(183,189),(184,190),(185,191),(186,192),(193,199),(194,200),(195,201),(196,202),(197,203),(198,204),(205,211),(206,212),(207,213),(208,214),(209,215),(210,216),(217,223),(218,224),(219,225),(220,226),(221,227),(222,228),(229,235),(230,236),(231,237),(232,238),(233,239),(234,240)], [(1,108),(2,97),(3,98),(4,99),(5,100),(6,101),(7,102),(8,103),(9,104),(10,105),(11,106),(12,107),(13,28),(14,29),(15,30),(16,31),(17,32),(18,33),(19,34),(20,35),(21,36),(22,25),(23,26),(24,27),(37,70),(38,71),(39,72),(40,61),(41,62),(42,63),(43,64),(44,65),(45,66),(46,67),(47,68),(48,69),(49,85),(50,86),(51,87),(52,88),(53,89),(54,90),(55,91),(56,92),(57,93),(58,94),(59,95),(60,96),(73,147),(74,148),(75,149),(76,150),(77,151),(78,152),(79,153),(80,154),(81,155),(82,156),(83,145),(84,146),(109,184),(110,185),(111,186),(112,187),(113,188),(114,189),(115,190),(116,191),(117,192),(118,181),(119,182),(120,183),(121,202),(122,203),(123,204),(124,193),(125,194),(126,195),(127,196),(128,197),(129,198),(130,199),(131,200),(132,201),(133,240),(134,229),(135,230),(136,231),(137,232),(138,233),(139,234),(140,235),(141,236),(142,237),(143,238),(144,239),(157,227),(158,228),(159,217),(160,218),(161,219),(162,220),(163,221),(164,222),(165,223),(166,224),(167,225),(168,226),(169,210),(170,211),(171,212),(172,213),(173,214),(174,215),(175,216),(176,205),(177,206),(178,207),(179,208),(180,209)], [(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)], [(1,6,7,12),(2,11,8,5),(3,4,9,10),(13,20,19,14),(15,18,21,24),(16,23,22,17),(25,32,31,26),(27,30,33,36),(28,35,34,29),(37,40,43,46),(38,45,44,39),(41,48,47,42),(49,60,55,54),(50,53,56,59),(51,58,57,52),(61,64,67,70),(62,69,68,63),(65,72,71,66),(73,76,79,82),(74,81,80,75),(77,84,83,78),(85,96,91,90),(86,89,92,95),(87,94,93,88),(97,106,103,100),(98,99,104,105),(101,102,107,108),(109,114,115,120),(110,119,116,113),(111,112,117,118),(121,126,127,132),(122,131,128,125),(123,124,129,130),(133,144,139,138),(134,137,140,143),(135,142,141,136),(145,152,151,146),(147,150,153,156),(148,155,154,149),(157,162,163,168),(158,167,164,161),(159,160,165,166),(169,178,175,172),(170,171,176,177),(173,174,179,180),(181,186,187,192),(182,191,188,185),(183,184,189,190),(193,198,199,204),(194,203,200,197),(195,196,201,202),(205,206,211,212),(207,216,213,210),(208,209,214,215),(217,218,223,224),(219,228,225,222),(220,221,226,227),(229,232,235,238),(230,237,236,231),(233,240,239,234)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 6E | 10A | ··· | 10L | 10M | 10N | 10O | 10P | 10Q | ··· | 10X | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 20M | ··· | 20T | 20U | ··· | 20AB | 30A | ··· | 30L | 30M | ··· | 30T | 60A | ··· | 60P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 2 | 2 | 2 | 4 | 6 | 6 | 12 | 12 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | C10 | S3 | D4 | D6 | D6 | C4○D4 | C5×S3 | C5×D4 | C4○D12 | S3×C10 | S3×C10 | C5×C4○D4 | C5×C4○D12 | S3×D4 | D4⋊2S3 | C5×S3×D4 | C5×D4⋊2S3 |
| kernel | C5×C23.9D6 | C5×Dic3⋊C4 | C5×C4⋊Dic3 | C5×D6⋊C4 | C5×C6.D4 | C15×C22⋊C4 | S3×C2×C20 | C10×C3⋊D4 | C23.9D6 | Dic3⋊C4 | C4⋊Dic3 | D6⋊C4 | C6.D4 | C3×C22⋊C4 | S3×C2×C4 | C2×C3⋊D4 | C5×C22⋊C4 | S3×C10 | C2×C20 | C22×C10 | C30 | C22⋊C4 | D6 | C10 | C2×C4 | C23 | C6 | C2 | C10 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 1 | 2 | 2 | 1 | 4 | 4 | 8 | 4 | 8 | 4 | 16 | 16 | 1 | 1 | 4 | 4 |
Matrix representation of C5×C23.9D6 ►in GL4(𝔽61) generated by
| 34 | 0 | 0 | 0 |
| 0 | 34 | 0 | 0 |
| 0 | 0 | 58 | 0 |
| 0 | 0 | 0 | 58 |
| 52 | 43 | 0 | 0 |
| 18 | 9 | 0 | 0 |
| 0 | 0 | 60 | 59 |
| 0 | 0 | 0 | 1 |
| 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 |
| 0 | 11 | 0 | 0 |
| 50 | 50 | 0 | 0 |
| 0 | 0 | 50 | 0 |
| 0 | 0 | 11 | 11 |
| 0 | 11 | 0 | 0 |
| 11 | 0 | 0 | 0 |
| 0 | 0 | 50 | 0 |
| 0 | 0 | 0 | 50 |
G:=sub<GL(4,GF(61))| [34,0,0,0,0,34,0,0,0,0,58,0,0,0,0,58],[52,18,0,0,43,9,0,0,0,0,60,0,0,0,59,1],[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],[0,50,0,0,11,50,0,0,0,0,50,11,0,0,0,11],[0,11,0,0,11,0,0,0,0,0,50,0,0,0,0,50] >;
C5×C23.9D6 in GAP, Magma, Sage, TeX
C_5\times C_2^3._9D_6
% in TeX
G:=Group("C5xC2^3.9D6"); // GroupNames label
G:=SmallGroup(480,762);
// by ID
G=gap.SmallGroup(480,762);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,288,926,891,15686]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=1,e^6=f^2=c,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=b*d=d*b,f*b*f^-1=b*c*d,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^5>;
// generators/relations