direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C23.14D6, (C6×D4)⋊9C10, (C2×C30)⋊21D4, D6⋊C4⋊15C10, (D4×C10)⋊12S3, (D4×C30)⋊25C2, Dic3⋊3(C5×D4), C6.51(D4×C10), C15⋊39(C4⋊D4), (C5×Dic3)⋊18D4, C10.194(S3×D4), C30.434(C2×D4), (C2×C20).244D6, Dic3⋊C4⋊15C10, C23.14(S3×C10), (C22×C10).94D6, C30.256(C4○D4), C6.D4⋊12C10, (C2×C30).433C23, (C2×C60).429C22, (C22×Dic3)⋊6C10, C10.123(D4⋊2S3), (C22×C30).126C22, (C10×Dic3).230C22, (C2×C6)⋊3(C5×D4), C3⋊5(C5×C4⋊D4), C2.27(C5×S3×D4), (C2×D4)⋊5(C5×S3), (C5×D6⋊C4)⋊37C2, (C2×C3⋊D4)⋊6C10, C6.31(C5×C4○D4), C22⋊2(C5×C3⋊D4), (C10×C3⋊D4)⋊21C2, (C2×C4).17(S3×C10), (Dic3×C2×C10)⋊17C2, C2.15(C10×C3⋊D4), C22.61(S3×C2×C10), (C2×C10)⋊11(C3⋊D4), (C2×C12).63(C2×C10), (C5×Dic3⋊C4)⋊37C2, C2.18(C5×D4⋊2S3), C10.136(C2×C3⋊D4), (S3×C2×C10).72C22, (C5×C6.D4)⋊28C2, (C22×C6).21(C2×C10), (C2×C6).54(C22×C10), (C22×S3).11(C2×C10), (C2×C10).367(C22×S3), (C2×Dic3).38(C2×C10), SmallGroup(480,818)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C23.14D6
G = < a,b,c,d,e,f | a5=b2=c2=d2=e6=1, f2=d, 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=de-1 >
Subgroups: 452 in 188 conjugacy classes, 70 normal (58 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, C10, C10, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C20, C2×C10, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C5×S3, C30, C30, C4⋊D4, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, Dic3⋊C4, D6⋊C4, C6.D4, C22×Dic3, C2×C3⋊D4, C6×D4, C5×Dic3, C5×Dic3, C60, S3×C10, C2×C30, C2×C30, C2×C30, C5×C22⋊C4, C5×C4⋊C4, C22×C20, D4×C10, D4×C10, C23.14D6, C10×Dic3, C10×Dic3, C5×C3⋊D4, C2×C60, D4×C15, S3×C2×C10, C22×C30, C5×C4⋊D4, C5×Dic3⋊C4, C5×D6⋊C4, C5×C6.D4, Dic3×C2×C10, C10×C3⋊D4, D4×C30, C5×C23.14D6
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, C2×D4, C4○D4, C2×C10, C3⋊D4, C22×S3, C5×S3, C4⋊D4, C5×D4, C22×C10, S3×D4, D4⋊2S3, C2×C3⋊D4, S3×C10, D4×C10, C5×C4○D4, C23.14D6, C5×C3⋊D4, S3×C2×C10, C5×C4⋊D4, C5×S3×D4, C5×D4⋊2S3, C10×C3⋊D4, C5×C23.14D6
►On 240 points
Generators in S240
(1 50 38 26 14)(2 51 39 27 15)(3 49 37 25 13)(4 52 40 28 16)(5 53 41 29 17)(6 54 42 30 18)(7 55 43 31 19)(8 56 44 32 20)(9 57 45 33 21)(10 58 46 34 22)(11 59 47 35 23)(12 60 48 36 24)(61 120 108 93 80)(62 118 106 91 81)(63 119 107 92 79)(64 110 69 94 83)(65 111 67 95 84)(66 109 68 96 82)(70 112 97 87 75)(71 113 98 85 73)(72 114 99 86 74)(76 115 103 101 88)(77 116 104 102 89)(78 117 105 100 90)(121 174 162 150 136)(122 169 157 145 137)(123 170 158 146 138)(124 171 159 147 133)(125 172 160 148 134)(126 173 161 149 135)(127 220 165 153 139)(128 221 166 154 140)(129 222 167 155 141)(130 217 168 156 142)(131 218 163 151 143)(132 219 164 152 144)(175 229 212 199 187)(176 230 213 200 188)(177 231 214 201 189)(178 232 215 202 190)(179 233 216 203 191)(180 234 211 204 192)(181 235 223 206 193)(182 236 224 207 194)(183 237 225 208 195)(184 238 226 209 196)(185 239 227 210 197)(186 240 228 205 198)
(1 217)(2 221)(3 219)(4 222)(5 220)(6 218)(7 123)(8 121)(9 125)(10 126)(11 124)(12 122)(13 132)(14 130)(15 128)(16 129)(17 127)(18 131)(19 138)(20 136)(21 134)(22 135)(23 133)(24 137)(25 144)(26 142)(27 140)(28 141)(29 139)(30 143)(31 146)(32 150)(33 148)(34 149)(35 147)(36 145)(37 152)(38 156)(39 154)(40 155)(41 153)(42 151)(43 158)(44 162)(45 160)(46 161)(47 159)(48 157)(49 164)(50 168)(51 166)(52 167)(53 165)(54 163)(55 170)(56 174)(57 172)(58 173)(59 171)(60 169)(61 181)(62 185)(63 183)(64 182)(65 186)(66 184)(67 228)(68 226)(69 224)(70 233)(71 231)(72 229)(73 177)(74 175)(75 179)(76 178)(77 176)(78 180)(79 195)(80 193)(81 197)(82 196)(83 194)(84 198)(85 189)(86 187)(87 191)(88 190)(89 188)(90 192)(91 210)(92 208)(93 206)(94 207)(95 205)(96 209)(97 203)(98 201)(99 199)(100 204)(101 202)(102 200)(103 215)(104 213)(105 211)(106 227)(107 225)(108 223)(109 238)(110 236)(111 240)(112 216)(113 214)(114 212)(115 232)(116 230)(117 234)(118 239)(119 237)(120 235)
(1 8)(2 9)(3 7)(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 74)(62 75)(63 73)(64 77)(65 78)(66 76)(67 105)(68 103)(69 104)(70 118)(71 119)(72 120)(79 85)(80 86)(81 87)(82 88)(83 89)(84 90)(91 97)(92 98)(93 99)(94 102)(95 100)(96 101)(106 112)(107 113)(108 114)(109 115)(110 116)(111 117)(121 217)(122 218)(123 219)(124 220)(125 221)(126 222)(127 133)(128 134)(129 135)(130 136)(131 137)(132 138)(139 147)(140 148)(141 149)(142 150)(143 145)(144 146)(151 157)(152 158)(153 159)(154 160)(155 161)(156 162)(163 169)(164 170)(165 171)(166 172)(167 173)(168 174)(175 181)(176 182)(177 183)(178 184)(179 185)(180 186)(187 193)(188 194)(189 195)(190 196)(191 197)(192 198)(199 206)(200 207)(201 208)(202 209)(203 210)(204 205)(211 228)(212 223)(213 224)(214 225)(215 226)(216 227)(229 235)(230 236)(231 237)(232 238)(233 239)(234 240)
(1 5)(2 6)(3 4)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(25 28)(26 29)(27 30)(31 34)(32 35)(33 36)(37 40)(38 41)(39 42)(43 46)(44 47)(45 48)(49 52)(50 53)(51 54)(55 58)(56 59)(57 60)(61 66)(62 64)(63 65)(67 107)(68 108)(69 106)(70 116)(71 117)(72 115)(73 78)(74 76)(75 77)(79 84)(80 82)(81 83)(85 90)(86 88)(87 89)(91 94)(92 95)(93 96)(97 102)(98 100)(99 101)(103 114)(104 112)(105 113)(109 120)(110 118)(111 119)(121 124)(122 125)(123 126)(127 130)(128 131)(129 132)(133 136)(134 137)(135 138)(139 142)(140 143)(141 144)(145 148)(146 149)(147 150)(151 154)(152 155)(153 156)(157 160)(158 161)(159 162)(163 166)(164 167)(165 168)(169 172)(170 173)(171 174)(175 178)(176 179)(177 180)(181 184)(182 185)(183 186)(187 190)(188 191)(189 192)(193 196)(194 197)(195 198)(199 202)(200 203)(201 204)(205 208)(206 209)(207 210)(211 214)(212 215)(213 216)(217 220)(218 221)(219 222)(223 226)(224 227)(225 228)(229 232)(230 233)(231 234)(235 238)(236 239)(237 240)
(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)
(1 179 5 176)(2 175 6 178)(3 177 4 180)(7 183 10 186)(8 185 11 182)(9 181 12 184)(13 189 16 192)(14 191 17 188)(15 187 18 190)(19 195 22 198)(20 197 23 194)(21 193 24 196)(25 201 28 204)(26 203 29 200)(27 199 30 202)(31 208 34 205)(32 210 35 207)(33 206 36 209)(37 214 40 211)(38 216 41 213)(39 212 42 215)(43 225 46 228)(44 227 47 224)(45 223 48 226)(49 231 52 234)(50 233 53 230)(51 229 54 232)(55 237 58 240)(56 239 59 236)(57 235 60 238)(61 221 66 218)(62 217 64 220)(63 219 65 222)(67 155 107 152)(68 151 108 154)(69 153 106 156)(70 174 116 171)(71 170 117 173)(72 172 115 169)(73 123 78 126)(74 125 76 122)(75 121 77 124)(79 132 84 129)(80 128 82 131)(81 130 83 127)(85 138 90 135)(86 134 88 137)(87 136 89 133)(91 142 94 139)(92 144 95 141)(93 140 96 143)(97 150 102 147)(98 146 100 149)(99 148 101 145)(103 157 114 160)(104 159 112 162)(105 161 113 158)(109 163 120 166)(110 165 118 168)(111 167 119 164)
G:=sub<Sym(240)| (1,50,38,26,14)(2,51,39,27,15)(3,49,37,25,13)(4,52,40,28,16)(5,53,41,29,17)(6,54,42,30,18)(7,55,43,31,19)(8,56,44,32,20)(9,57,45,33,21)(10,58,46,34,22)(11,59,47,35,23)(12,60,48,36,24)(61,120,108,93,80)(62,118,106,91,81)(63,119,107,92,79)(64,110,69,94,83)(65,111,67,95,84)(66,109,68,96,82)(70,112,97,87,75)(71,113,98,85,73)(72,114,99,86,74)(76,115,103,101,88)(77,116,104,102,89)(78,117,105,100,90)(121,174,162,150,136)(122,169,157,145,137)(123,170,158,146,138)(124,171,159,147,133)(125,172,160,148,134)(126,173,161,149,135)(127,220,165,153,139)(128,221,166,154,140)(129,222,167,155,141)(130,217,168,156,142)(131,218,163,151,143)(132,219,164,152,144)(175,229,212,199,187)(176,230,213,200,188)(177,231,214,201,189)(178,232,215,202,190)(179,233,216,203,191)(180,234,211,204,192)(181,235,223,206,193)(182,236,224,207,194)(183,237,225,208,195)(184,238,226,209,196)(185,239,227,210,197)(186,240,228,205,198), (1,217)(2,221)(3,219)(4,222)(5,220)(6,218)(7,123)(8,121)(9,125)(10,126)(11,124)(12,122)(13,132)(14,130)(15,128)(16,129)(17,127)(18,131)(19,138)(20,136)(21,134)(22,135)(23,133)(24,137)(25,144)(26,142)(27,140)(28,141)(29,139)(30,143)(31,146)(32,150)(33,148)(34,149)(35,147)(36,145)(37,152)(38,156)(39,154)(40,155)(41,153)(42,151)(43,158)(44,162)(45,160)(46,161)(47,159)(48,157)(49,164)(50,168)(51,166)(52,167)(53,165)(54,163)(55,170)(56,174)(57,172)(58,173)(59,171)(60,169)(61,181)(62,185)(63,183)(64,182)(65,186)(66,184)(67,228)(68,226)(69,224)(70,233)(71,231)(72,229)(73,177)(74,175)(75,179)(76,178)(77,176)(78,180)(79,195)(80,193)(81,197)(82,196)(83,194)(84,198)(85,189)(86,187)(87,191)(88,190)(89,188)(90,192)(91,210)(92,208)(93,206)(94,207)(95,205)(96,209)(97,203)(98,201)(99,199)(100,204)(101,202)(102,200)(103,215)(104,213)(105,211)(106,227)(107,225)(108,223)(109,238)(110,236)(111,240)(112,216)(113,214)(114,212)(115,232)(116,230)(117,234)(118,239)(119,237)(120,235), (1,8)(2,9)(3,7)(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)(61,74)(62,75)(63,73)(64,77)(65,78)(66,76)(67,105)(68,103)(69,104)(70,118)(71,119)(72,120)(79,85)(80,86)(81,87)(82,88)(83,89)(84,90)(91,97)(92,98)(93,99)(94,102)(95,100)(96,101)(106,112)(107,113)(108,114)(109,115)(110,116)(111,117)(121,217)(122,218)(123,219)(124,220)(125,221)(126,222)(127,133)(128,134)(129,135)(130,136)(131,137)(132,138)(139,147)(140,148)(141,149)(142,150)(143,145)(144,146)(151,157)(152,158)(153,159)(154,160)(155,161)(156,162)(163,169)(164,170)(165,171)(166,172)(167,173)(168,174)(175,181)(176,182)(177,183)(178,184)(179,185)(180,186)(187,193)(188,194)(189,195)(190,196)(191,197)(192,198)(199,206)(200,207)(201,208)(202,209)(203,210)(204,205)(211,228)(212,223)(213,224)(214,225)(215,226)(216,227)(229,235)(230,236)(231,237)(232,238)(233,239)(234,240), (1,5)(2,6)(3,4)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,34)(32,35)(33,36)(37,40)(38,41)(39,42)(43,46)(44,47)(45,48)(49,52)(50,53)(51,54)(55,58)(56,59)(57,60)(61,66)(62,64)(63,65)(67,107)(68,108)(69,106)(70,116)(71,117)(72,115)(73,78)(74,76)(75,77)(79,84)(80,82)(81,83)(85,90)(86,88)(87,89)(91,94)(92,95)(93,96)(97,102)(98,100)(99,101)(103,114)(104,112)(105,113)(109,120)(110,118)(111,119)(121,124)(122,125)(123,126)(127,130)(128,131)(129,132)(133,136)(134,137)(135,138)(139,142)(140,143)(141,144)(145,148)(146,149)(147,150)(151,154)(152,155)(153,156)(157,160)(158,161)(159,162)(163,166)(164,167)(165,168)(169,172)(170,173)(171,174)(175,178)(176,179)(177,180)(181,184)(182,185)(183,186)(187,190)(188,191)(189,192)(193,196)(194,197)(195,198)(199,202)(200,203)(201,204)(205,208)(206,209)(207,210)(211,214)(212,215)(213,216)(217,220)(218,221)(219,222)(223,226)(224,227)(225,228)(229,232)(230,233)(231,234)(235,238)(236,239)(237,240), (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), (1,179,5,176)(2,175,6,178)(3,177,4,180)(7,183,10,186)(8,185,11,182)(9,181,12,184)(13,189,16,192)(14,191,17,188)(15,187,18,190)(19,195,22,198)(20,197,23,194)(21,193,24,196)(25,201,28,204)(26,203,29,200)(27,199,30,202)(31,208,34,205)(32,210,35,207)(33,206,36,209)(37,214,40,211)(38,216,41,213)(39,212,42,215)(43,225,46,228)(44,227,47,224)(45,223,48,226)(49,231,52,234)(50,233,53,230)(51,229,54,232)(55,237,58,240)(56,239,59,236)(57,235,60,238)(61,221,66,218)(62,217,64,220)(63,219,65,222)(67,155,107,152)(68,151,108,154)(69,153,106,156)(70,174,116,171)(71,170,117,173)(72,172,115,169)(73,123,78,126)(74,125,76,122)(75,121,77,124)(79,132,84,129)(80,128,82,131)(81,130,83,127)(85,138,90,135)(86,134,88,137)(87,136,89,133)(91,142,94,139)(92,144,95,141)(93,140,96,143)(97,150,102,147)(98,146,100,149)(99,148,101,145)(103,157,114,160)(104,159,112,162)(105,161,113,158)(109,163,120,166)(110,165,118,168)(111,167,119,164)>;
G:=Group( (1,50,38,26,14)(2,51,39,27,15)(3,49,37,25,13)(4,52,40,28,16)(5,53,41,29,17)(6,54,42,30,18)(7,55,43,31,19)(8,56,44,32,20)(9,57,45,33,21)(10,58,46,34,22)(11,59,47,35,23)(12,60,48,36,24)(61,120,108,93,80)(62,118,106,91,81)(63,119,107,92,79)(64,110,69,94,83)(65,111,67,95,84)(66,109,68,96,82)(70,112,97,87,75)(71,113,98,85,73)(72,114,99,86,74)(76,115,103,101,88)(77,116,104,102,89)(78,117,105,100,90)(121,174,162,150,136)(122,169,157,145,137)(123,170,158,146,138)(124,171,159,147,133)(125,172,160,148,134)(126,173,161,149,135)(127,220,165,153,139)(128,221,166,154,140)(129,222,167,155,141)(130,217,168,156,142)(131,218,163,151,143)(132,219,164,152,144)(175,229,212,199,187)(176,230,213,200,188)(177,231,214,201,189)(178,232,215,202,190)(179,233,216,203,191)(180,234,211,204,192)(181,235,223,206,193)(182,236,224,207,194)(183,237,225,208,195)(184,238,226,209,196)(185,239,227,210,197)(186,240,228,205,198), (1,217)(2,221)(3,219)(4,222)(5,220)(6,218)(7,123)(8,121)(9,125)(10,126)(11,124)(12,122)(13,132)(14,130)(15,128)(16,129)(17,127)(18,131)(19,138)(20,136)(21,134)(22,135)(23,133)(24,137)(25,144)(26,142)(27,140)(28,141)(29,139)(30,143)(31,146)(32,150)(33,148)(34,149)(35,147)(36,145)(37,152)(38,156)(39,154)(40,155)(41,153)(42,151)(43,158)(44,162)(45,160)(46,161)(47,159)(48,157)(49,164)(50,168)(51,166)(52,167)(53,165)(54,163)(55,170)(56,174)(57,172)(58,173)(59,171)(60,169)(61,181)(62,185)(63,183)(64,182)(65,186)(66,184)(67,228)(68,226)(69,224)(70,233)(71,231)(72,229)(73,177)(74,175)(75,179)(76,178)(77,176)(78,180)(79,195)(80,193)(81,197)(82,196)(83,194)(84,198)(85,189)(86,187)(87,191)(88,190)(89,188)(90,192)(91,210)(92,208)(93,206)(94,207)(95,205)(96,209)(97,203)(98,201)(99,199)(100,204)(101,202)(102,200)(103,215)(104,213)(105,211)(106,227)(107,225)(108,223)(109,238)(110,236)(111,240)(112,216)(113,214)(114,212)(115,232)(116,230)(117,234)(118,239)(119,237)(120,235), (1,8)(2,9)(3,7)(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)(61,74)(62,75)(63,73)(64,77)(65,78)(66,76)(67,105)(68,103)(69,104)(70,118)(71,119)(72,120)(79,85)(80,86)(81,87)(82,88)(83,89)(84,90)(91,97)(92,98)(93,99)(94,102)(95,100)(96,101)(106,112)(107,113)(108,114)(109,115)(110,116)(111,117)(121,217)(122,218)(123,219)(124,220)(125,221)(126,222)(127,133)(128,134)(129,135)(130,136)(131,137)(132,138)(139,147)(140,148)(141,149)(142,150)(143,145)(144,146)(151,157)(152,158)(153,159)(154,160)(155,161)(156,162)(163,169)(164,170)(165,171)(166,172)(167,173)(168,174)(175,181)(176,182)(177,183)(178,184)(179,185)(180,186)(187,193)(188,194)(189,195)(190,196)(191,197)(192,198)(199,206)(200,207)(201,208)(202,209)(203,210)(204,205)(211,228)(212,223)(213,224)(214,225)(215,226)(216,227)(229,235)(230,236)(231,237)(232,238)(233,239)(234,240), (1,5)(2,6)(3,4)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,34)(32,35)(33,36)(37,40)(38,41)(39,42)(43,46)(44,47)(45,48)(49,52)(50,53)(51,54)(55,58)(56,59)(57,60)(61,66)(62,64)(63,65)(67,107)(68,108)(69,106)(70,116)(71,117)(72,115)(73,78)(74,76)(75,77)(79,84)(80,82)(81,83)(85,90)(86,88)(87,89)(91,94)(92,95)(93,96)(97,102)(98,100)(99,101)(103,114)(104,112)(105,113)(109,120)(110,118)(111,119)(121,124)(122,125)(123,126)(127,130)(128,131)(129,132)(133,136)(134,137)(135,138)(139,142)(140,143)(141,144)(145,148)(146,149)(147,150)(151,154)(152,155)(153,156)(157,160)(158,161)(159,162)(163,166)(164,167)(165,168)(169,172)(170,173)(171,174)(175,178)(176,179)(177,180)(181,184)(182,185)(183,186)(187,190)(188,191)(189,192)(193,196)(194,197)(195,198)(199,202)(200,203)(201,204)(205,208)(206,209)(207,210)(211,214)(212,215)(213,216)(217,220)(218,221)(219,222)(223,226)(224,227)(225,228)(229,232)(230,233)(231,234)(235,238)(236,239)(237,240), (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), (1,179,5,176)(2,175,6,178)(3,177,4,180)(7,183,10,186)(8,185,11,182)(9,181,12,184)(13,189,16,192)(14,191,17,188)(15,187,18,190)(19,195,22,198)(20,197,23,194)(21,193,24,196)(25,201,28,204)(26,203,29,200)(27,199,30,202)(31,208,34,205)(32,210,35,207)(33,206,36,209)(37,214,40,211)(38,216,41,213)(39,212,42,215)(43,225,46,228)(44,227,47,224)(45,223,48,226)(49,231,52,234)(50,233,53,230)(51,229,54,232)(55,237,58,240)(56,239,59,236)(57,235,60,238)(61,221,66,218)(62,217,64,220)(63,219,65,222)(67,155,107,152)(68,151,108,154)(69,153,106,156)(70,174,116,171)(71,170,117,173)(72,172,115,169)(73,123,78,126)(74,125,76,122)(75,121,77,124)(79,132,84,129)(80,128,82,131)(81,130,83,127)(85,138,90,135)(86,134,88,137)(87,136,89,133)(91,142,94,139)(92,144,95,141)(93,140,96,143)(97,150,102,147)(98,146,100,149)(99,148,101,145)(103,157,114,160)(104,159,112,162)(105,161,113,158)(109,163,120,166)(110,165,118,168)(111,167,119,164) );
G=PermutationGroup([[(1,50,38,26,14),(2,51,39,27,15),(3,49,37,25,13),(4,52,40,28,16),(5,53,41,29,17),(6,54,42,30,18),(7,55,43,31,19),(8,56,44,32,20),(9,57,45,33,21),(10,58,46,34,22),(11,59,47,35,23),(12,60,48,36,24),(61,120,108,93,80),(62,118,106,91,81),(63,119,107,92,79),(64,110,69,94,83),(65,111,67,95,84),(66,109,68,96,82),(70,112,97,87,75),(71,113,98,85,73),(72,114,99,86,74),(76,115,103,101,88),(77,116,104,102,89),(78,117,105,100,90),(121,174,162,150,136),(122,169,157,145,137),(123,170,158,146,138),(124,171,159,147,133),(125,172,160,148,134),(126,173,161,149,135),(127,220,165,153,139),(128,221,166,154,140),(129,222,167,155,141),(130,217,168,156,142),(131,218,163,151,143),(132,219,164,152,144),(175,229,212,199,187),(176,230,213,200,188),(177,231,214,201,189),(178,232,215,202,190),(179,233,216,203,191),(180,234,211,204,192),(181,235,223,206,193),(182,236,224,207,194),(183,237,225,208,195),(184,238,226,209,196),(185,239,227,210,197),(186,240,228,205,198)], [(1,217),(2,221),(3,219),(4,222),(5,220),(6,218),(7,123),(8,121),(9,125),(10,126),(11,124),(12,122),(13,132),(14,130),(15,128),(16,129),(17,127),(18,131),(19,138),(20,136),(21,134),(22,135),(23,133),(24,137),(25,144),(26,142),(27,140),(28,141),(29,139),(30,143),(31,146),(32,150),(33,148),(34,149),(35,147),(36,145),(37,152),(38,156),(39,154),(40,155),(41,153),(42,151),(43,158),(44,162),(45,160),(46,161),(47,159),(48,157),(49,164),(50,168),(51,166),(52,167),(53,165),(54,163),(55,170),(56,174),(57,172),(58,173),(59,171),(60,169),(61,181),(62,185),(63,183),(64,182),(65,186),(66,184),(67,228),(68,226),(69,224),(70,233),(71,231),(72,229),(73,177),(74,175),(75,179),(76,178),(77,176),(78,180),(79,195),(80,193),(81,197),(82,196),(83,194),(84,198),(85,189),(86,187),(87,191),(88,190),(89,188),(90,192),(91,210),(92,208),(93,206),(94,207),(95,205),(96,209),(97,203),(98,201),(99,199),(100,204),(101,202),(102,200),(103,215),(104,213),(105,211),(106,227),(107,225),(108,223),(109,238),(110,236),(111,240),(112,216),(113,214),(114,212),(115,232),(116,230),(117,234),(118,239),(119,237),(120,235)], [(1,8),(2,9),(3,7),(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),(61,74),(62,75),(63,73),(64,77),(65,78),(66,76),(67,105),(68,103),(69,104),(70,118),(71,119),(72,120),(79,85),(80,86),(81,87),(82,88),(83,89),(84,90),(91,97),(92,98),(93,99),(94,102),(95,100),(96,101),(106,112),(107,113),(108,114),(109,115),(110,116),(111,117),(121,217),(122,218),(123,219),(124,220),(125,221),(126,222),(127,133),(128,134),(129,135),(130,136),(131,137),(132,138),(139,147),(140,148),(141,149),(142,150),(143,145),(144,146),(151,157),(152,158),(153,159),(154,160),(155,161),(156,162),(163,169),(164,170),(165,171),(166,172),(167,173),(168,174),(175,181),(176,182),(177,183),(178,184),(179,185),(180,186),(187,193),(188,194),(189,195),(190,196),(191,197),(192,198),(199,206),(200,207),(201,208),(202,209),(203,210),(204,205),(211,228),(212,223),(213,224),(214,225),(215,226),(216,227),(229,235),(230,236),(231,237),(232,238),(233,239),(234,240)], [(1,5),(2,6),(3,4),(7,10),(8,11),(9,12),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24),(25,28),(26,29),(27,30),(31,34),(32,35),(33,36),(37,40),(38,41),(39,42),(43,46),(44,47),(45,48),(49,52),(50,53),(51,54),(55,58),(56,59),(57,60),(61,66),(62,64),(63,65),(67,107),(68,108),(69,106),(70,116),(71,117),(72,115),(73,78),(74,76),(75,77),(79,84),(80,82),(81,83),(85,90),(86,88),(87,89),(91,94),(92,95),(93,96),(97,102),(98,100),(99,101),(103,114),(104,112),(105,113),(109,120),(110,118),(111,119),(121,124),(122,125),(123,126),(127,130),(128,131),(129,132),(133,136),(134,137),(135,138),(139,142),(140,143),(141,144),(145,148),(146,149),(147,150),(151,154),(152,155),(153,156),(157,160),(158,161),(159,162),(163,166),(164,167),(165,168),(169,172),(170,173),(171,174),(175,178),(176,179),(177,180),(181,184),(182,185),(183,186),(187,190),(188,191),(189,192),(193,196),(194,197),(195,198),(199,202),(200,203),(201,204),(205,208),(206,209),(207,210),(211,214),(212,215),(213,216),(217,220),(218,221),(219,222),(223,226),(224,227),(225,228),(229,232),(230,233),(231,234),(235,238),(236,239),(237,240)], [(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)], [(1,179,5,176),(2,175,6,178),(3,177,4,180),(7,183,10,186),(8,185,11,182),(9,181,12,184),(13,189,16,192),(14,191,17,188),(15,187,18,190),(19,195,22,198),(20,197,23,194),(21,193,24,196),(25,201,28,204),(26,203,29,200),(27,199,30,202),(31,208,34,205),(32,210,35,207),(33,206,36,209),(37,214,40,211),(38,216,41,213),(39,212,42,215),(43,225,46,228),(44,227,47,224),(45,223,48,226),(49,231,52,234),(50,233,53,230),(51,229,54,232),(55,237,58,240),(56,239,59,236),(57,235,60,238),(61,221,66,218),(62,217,64,220),(63,219,65,222),(67,155,107,152),(68,151,108,154),(69,153,106,156),(70,174,116,171),(71,170,117,173),(72,172,115,169),(73,123,78,126),(74,125,76,122),(75,121,77,124),(79,132,84,129),(80,128,82,131),(81,130,83,127),(85,138,90,135),(86,134,88,137),(87,136,89,133),(91,142,94,139),(92,144,95,141),(93,140,96,143),(97,150,102,147),(98,146,100,149),(99,148,101,145),(103,157,114,160),(104,159,112,162),(105,161,113,158),(109,163,120,166),(110,165,118,168),(111,167,119,164)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10L | 10M | ··· | 10T | 10U | 10V | 10W | 10X | 10Y | 10Z | 10AA | 10AB | 12A | 12B | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | ··· | 20T | 20U | 20V | 20W | 20X | 30A | ··· | 30L | 30M | ··· | 30AB | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 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 | 2 | 2 | 4 | 12 | 2 | 4 | 6 | 6 | 6 | 6 | 12 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 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 | 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 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | S3 | D4 | D4 | D6 | D6 | C4○D4 | C3⋊D4 | C5×S3 | C5×D4 | C5×D4 | S3×C10 | S3×C10 | C5×C4○D4 | C5×C3⋊D4 | S3×D4 | D4⋊2S3 | C5×S3×D4 | C5×D4⋊2S3 |
| kernel | C5×C23.14D6 | C5×Dic3⋊C4 | C5×D6⋊C4 | C5×C6.D4 | Dic3×C2×C10 | C10×C3⋊D4 | D4×C30 | C23.14D6 | Dic3⋊C4 | D6⋊C4 | C6.D4 | C22×Dic3 | C2×C3⋊D4 | C6×D4 | D4×C10 | C5×Dic3 | C2×C30 | C2×C20 | C22×C10 | C30 | C2×C10 | C2×D4 | Dic3 | C2×C6 | C2×C4 | C23 | C6 | C22 | C10 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 4 | 4 | 4 | 4 | 4 | 8 | 4 | 1 | 2 | 2 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 4 | 8 | 8 | 16 | 1 | 1 | 4 | 4 |
Matrix representation of C5×C23.14D6 ►in GL6(𝔽61)
| 34 | 0 | 0 | 0 | 0 | 0 |
| 0 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 58 | 0 | 0 | 0 |
| 0 | 0 | 0 | 58 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 40 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 60 | 1 | 0 | 0 | 0 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 3 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 48 | 22 | 0 | 0 | 0 | 0 |
| 9 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 13 | 0 | 0 |
| 0 | 0 | 0 | 50 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 60 | 0 |
G:=sub<GL(6,GF(61))| [34,0,0,0,0,0,0,34,0,0,0,0,0,0,58,0,0,0,0,0,0,58,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,40,60,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[60,60,0,0,0,0,1,0,0,0,0,0,0,0,1,3,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,60],[48,9,0,0,0,0,22,13,0,0,0,0,0,0,11,0,0,0,0,0,13,50,0,0,0,0,0,0,0,60,0,0,0,0,1,0] >;
C5×C23.14D6 in GAP, Magma, Sage, TeX
C_5\times C_2^3._{14}D_6 % in TeX
G:=Group("C5xC2^3.14D6"); // GroupNames label
G:=SmallGroup(480,818);
// by ID
G=gap.SmallGroup(480,818);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1149,926,891,15686]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=e^6=1,f^2=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=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=d*e^-1>;
// generators/relations