direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: S3×C4⋊Dic5, D6.13D20, D6.2Dic10, (S3×C20)⋊6C4, C20⋊17(C4×S3), C60⋊15(C2×C4), C4⋊3(S3×Dic5), C2.3(S3×D20), (C4×S3)⋊2Dic5, C60⋊5C4⋊26C2, C12⋊1(C2×Dic5), C10.16(S3×D4), C6.16(C2×D20), C30.47(C2×D4), (S3×C10).2Q8, C10.34(S3×Q8), C30.45(C2×Q8), (S3×C10).23D4, (C2×C20).299D6, D6.9(C2×Dic5), C2.5(S3×Dic10), Dic3⋊3(C2×Dic5), (C2×C12).125D10, C6.Dic10⋊20C2, C6.16(C2×Dic10), (C2×C30).116C23, (C2×C60).118C22, C30.128(C22×C4), (C2×Dic5).110D6, (C22×S3).86D10, C6.12(C22×Dic5), (C2×Dic3).150D10, (C6×Dic5).69C22, (C2×Dic15).94C22, (C10×Dic3).181C22, C5⋊6(S3×C4⋊C4), C15⋊9(C2×C4⋊C4), C3⋊1(C2×C4⋊Dic5), (S3×C2×C4).5D5, (S3×C2×C20).5C2, (C5×S3)⋊3(C4⋊C4), C10.119(S3×C2×C4), (C3×C4⋊Dic5)⋊4C2, (C2×S3×Dic5).6C2, C2.13(C2×S3×Dic5), C22.56(C2×S3×D5), (C2×C4).109(S3×D5), (S3×C10).38(C2×C4), (C5×Dic3)⋊20(C2×C4), (S3×C2×C10).83C22, (C2×C6).128(C22×D5), (C2×C10).128(C22×S3), SmallGroup(480,502)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×C4⋊Dic5
G = < a,b,c,d,e | a3=b2=c4=d10=1, e2=d5, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >
Subgroups: 748 in 184 conjugacy classes, 84 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, C23, C10, C10, Dic3, Dic3, C12, C12, D6, C2×C6, C15, C4⋊C4, C22×C4, Dic5, C20, C20, C2×C10, C2×C10, C4×S3, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C5×S3, C30, C2×C4⋊C4, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, S3×C2×C4, S3×C2×C4, C5×Dic3, C3×Dic5, Dic15, C60, S3×C10, C2×C30, C4⋊Dic5, C4⋊Dic5, C22×Dic5, C22×C20, S3×C4⋊C4, S3×Dic5, C6×Dic5, S3×C20, C10×Dic3, C2×Dic15, C2×C60, S3×C2×C10, C2×C4⋊Dic5, C6.Dic10, C3×C4⋊Dic5, C60⋊5C4, C2×S3×Dic5, S3×C2×C20, S3×C4⋊Dic5
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D5, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic5, D10, C4×S3, C22×S3, C2×C4⋊C4, Dic10, D20, C2×Dic5, C22×D5, S3×C2×C4, S3×D4, S3×Q8, S3×D5, C4⋊Dic5, C2×Dic10, C2×D20, C22×Dic5, S3×C4⋊C4, S3×Dic5, C2×S3×D5, C2×C4⋊Dic5, S3×Dic10, S3×D20, C2×S3×Dic5, S3×C4⋊Dic5
►On 240 points
Generators in S240
(1 27 45)(2 28 46)(3 29 47)(4 30 48)(5 21 49)(6 22 50)(7 23 41)(8 24 42)(9 25 43)(10 26 44)(11 67 51)(12 68 52)(13 69 53)(14 70 54)(15 61 55)(16 62 56)(17 63 57)(18 64 58)(19 65 59)(20 66 60)(31 239 220)(32 240 211)(33 231 212)(34 232 213)(35 233 214)(36 234 215)(37 235 216)(38 236 217)(39 237 218)(40 238 219)(71 87 105)(72 88 106)(73 89 107)(74 90 108)(75 81 109)(76 82 110)(77 83 101)(78 84 102)(79 85 103)(80 86 104)(91 121 112)(92 122 113)(93 123 114)(94 124 115)(95 125 116)(96 126 117)(97 127 118)(98 128 119)(99 129 120)(100 130 111)(131 145 162)(132 146 163)(133 147 164)(134 148 165)(135 149 166)(136 150 167)(137 141 168)(138 142 169)(139 143 170)(140 144 161)(151 189 180)(152 190 171)(153 181 172)(154 182 173)(155 183 174)(156 184 175)(157 185 176)(158 186 177)(159 187 178)(160 188 179)(191 205 222)(192 206 223)(193 207 224)(194 208 225)(195 209 226)(196 210 227)(197 201 228)(198 202 229)(199 203 230)(200 204 221)
(11 67)(12 68)(13 69)(14 70)(15 61)(16 62)(17 63)(18 64)(19 65)(20 66)(21 49)(22 50)(23 41)(24 42)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 220)(32 211)(33 212)(34 213)(35 214)(36 215)(37 216)(38 217)(39 218)(40 219)(81 109)(82 110)(83 101)(84 102)(85 103)(86 104)(87 105)(88 106)(89 107)(90 108)(91 121)(92 122)(93 123)(94 124)(95 125)(96 126)(97 127)(98 128)(99 129)(100 130)(141 168)(142 169)(143 170)(144 161)(145 162)(146 163)(147 164)(148 165)(149 166)(150 167)(151 189)(152 190)(153 181)(154 182)(155 183)(156 184)(157 185)(158 186)(159 187)(160 188)(201 228)(202 229)(203 230)(204 221)(205 222)(206 223)(207 224)(208 225)(209 226)(210 227)
(1 111 51 71)(2 112 52 72)(3 113 53 73)(4 114 54 74)(5 115 55 75)(6 116 56 76)(7 117 57 77)(8 118 58 78)(9 119 59 79)(10 120 60 80)(11 87 27 100)(12 88 28 91)(13 89 29 92)(14 90 30 93)(15 81 21 94)(16 82 22 95)(17 83 23 96)(18 84 24 97)(19 85 25 98)(20 86 26 99)(31 165 225 188)(32 166 226 189)(33 167 227 190)(34 168 228 181)(35 169 229 182)(36 170 230 183)(37 161 221 184)(38 162 222 185)(39 163 223 186)(40 164 224 187)(41 126 63 101)(42 127 64 102)(43 128 65 103)(44 129 66 104)(45 130 67 105)(46 121 68 106)(47 122 69 107)(48 123 70 108)(49 124 61 109)(50 125 62 110)(131 191 176 236)(132 192 177 237)(133 193 178 238)(134 194 179 239)(135 195 180 240)(136 196 171 231)(137 197 172 232)(138 198 173 233)(139 199 174 234)(140 200 175 235)(141 201 153 213)(142 202 154 214)(143 203 155 215)(144 204 156 216)(145 205 157 217)(146 206 158 218)(147 207 159 219)(148 208 160 220)(149 209 151 211)(150 210 152 212)
(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)
(1 139 6 134)(2 138 7 133)(3 137 8 132)(4 136 9 131)(5 135 10 140)(11 155 16 160)(12 154 17 159)(13 153 18 158)(14 152 19 157)(15 151 20 156)(21 149 26 144)(22 148 27 143)(23 147 28 142)(24 146 29 141)(25 145 30 150)(31 130 36 125)(32 129 37 124)(33 128 38 123)(34 127 39 122)(35 126 40 121)(41 164 46 169)(42 163 47 168)(43 162 48 167)(44 161 49 166)(45 170 50 165)(51 174 56 179)(52 173 57 178)(53 172 58 177)(54 171 59 176)(55 180 60 175)(61 189 66 184)(62 188 67 183)(63 187 68 182)(64 186 69 181)(65 185 70 190)(71 199 76 194)(72 198 77 193)(73 197 78 192)(74 196 79 191)(75 195 80 200)(81 209 86 204)(82 208 87 203)(83 207 88 202)(84 206 89 201)(85 205 90 210)(91 214 96 219)(92 213 97 218)(93 212 98 217)(94 211 99 216)(95 220 100 215)(101 224 106 229)(102 223 107 228)(103 222 108 227)(104 221 109 226)(105 230 110 225)(111 234 116 239)(112 233 117 238)(113 232 118 237)(114 231 119 236)(115 240 120 235)
G:=sub<Sym(240)| (1,27,45)(2,28,46)(3,29,47)(4,30,48)(5,21,49)(6,22,50)(7,23,41)(8,24,42)(9,25,43)(10,26,44)(11,67,51)(12,68,52)(13,69,53)(14,70,54)(15,61,55)(16,62,56)(17,63,57)(18,64,58)(19,65,59)(20,66,60)(31,239,220)(32,240,211)(33,231,212)(34,232,213)(35,233,214)(36,234,215)(37,235,216)(38,236,217)(39,237,218)(40,238,219)(71,87,105)(72,88,106)(73,89,107)(74,90,108)(75,81,109)(76,82,110)(77,83,101)(78,84,102)(79,85,103)(80,86,104)(91,121,112)(92,122,113)(93,123,114)(94,124,115)(95,125,116)(96,126,117)(97,127,118)(98,128,119)(99,129,120)(100,130,111)(131,145,162)(132,146,163)(133,147,164)(134,148,165)(135,149,166)(136,150,167)(137,141,168)(138,142,169)(139,143,170)(140,144,161)(151,189,180)(152,190,171)(153,181,172)(154,182,173)(155,183,174)(156,184,175)(157,185,176)(158,186,177)(159,187,178)(160,188,179)(191,205,222)(192,206,223)(193,207,224)(194,208,225)(195,209,226)(196,210,227)(197,201,228)(198,202,229)(199,203,230)(200,204,221), (11,67)(12,68)(13,69)(14,70)(15,61)(16,62)(17,63)(18,64)(19,65)(20,66)(21,49)(22,50)(23,41)(24,42)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,220)(32,211)(33,212)(34,213)(35,214)(36,215)(37,216)(38,217)(39,218)(40,219)(81,109)(82,110)(83,101)(84,102)(85,103)(86,104)(87,105)(88,106)(89,107)(90,108)(91,121)(92,122)(93,123)(94,124)(95,125)(96,126)(97,127)(98,128)(99,129)(100,130)(141,168)(142,169)(143,170)(144,161)(145,162)(146,163)(147,164)(148,165)(149,166)(150,167)(151,189)(152,190)(153,181)(154,182)(155,183)(156,184)(157,185)(158,186)(159,187)(160,188)(201,228)(202,229)(203,230)(204,221)(205,222)(206,223)(207,224)(208,225)(209,226)(210,227), (1,111,51,71)(2,112,52,72)(3,113,53,73)(4,114,54,74)(5,115,55,75)(6,116,56,76)(7,117,57,77)(8,118,58,78)(9,119,59,79)(10,120,60,80)(11,87,27,100)(12,88,28,91)(13,89,29,92)(14,90,30,93)(15,81,21,94)(16,82,22,95)(17,83,23,96)(18,84,24,97)(19,85,25,98)(20,86,26,99)(31,165,225,188)(32,166,226,189)(33,167,227,190)(34,168,228,181)(35,169,229,182)(36,170,230,183)(37,161,221,184)(38,162,222,185)(39,163,223,186)(40,164,224,187)(41,126,63,101)(42,127,64,102)(43,128,65,103)(44,129,66,104)(45,130,67,105)(46,121,68,106)(47,122,69,107)(48,123,70,108)(49,124,61,109)(50,125,62,110)(131,191,176,236)(132,192,177,237)(133,193,178,238)(134,194,179,239)(135,195,180,240)(136,196,171,231)(137,197,172,232)(138,198,173,233)(139,199,174,234)(140,200,175,235)(141,201,153,213)(142,202,154,214)(143,203,155,215)(144,204,156,216)(145,205,157,217)(146,206,158,218)(147,207,159,219)(148,208,160,220)(149,209,151,211)(150,210,152,212), (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), (1,139,6,134)(2,138,7,133)(3,137,8,132)(4,136,9,131)(5,135,10,140)(11,155,16,160)(12,154,17,159)(13,153,18,158)(14,152,19,157)(15,151,20,156)(21,149,26,144)(22,148,27,143)(23,147,28,142)(24,146,29,141)(25,145,30,150)(31,130,36,125)(32,129,37,124)(33,128,38,123)(34,127,39,122)(35,126,40,121)(41,164,46,169)(42,163,47,168)(43,162,48,167)(44,161,49,166)(45,170,50,165)(51,174,56,179)(52,173,57,178)(53,172,58,177)(54,171,59,176)(55,180,60,175)(61,189,66,184)(62,188,67,183)(63,187,68,182)(64,186,69,181)(65,185,70,190)(71,199,76,194)(72,198,77,193)(73,197,78,192)(74,196,79,191)(75,195,80,200)(81,209,86,204)(82,208,87,203)(83,207,88,202)(84,206,89,201)(85,205,90,210)(91,214,96,219)(92,213,97,218)(93,212,98,217)(94,211,99,216)(95,220,100,215)(101,224,106,229)(102,223,107,228)(103,222,108,227)(104,221,109,226)(105,230,110,225)(111,234,116,239)(112,233,117,238)(113,232,118,237)(114,231,119,236)(115,240,120,235)>;
G:=Group( (1,27,45)(2,28,46)(3,29,47)(4,30,48)(5,21,49)(6,22,50)(7,23,41)(8,24,42)(9,25,43)(10,26,44)(11,67,51)(12,68,52)(13,69,53)(14,70,54)(15,61,55)(16,62,56)(17,63,57)(18,64,58)(19,65,59)(20,66,60)(31,239,220)(32,240,211)(33,231,212)(34,232,213)(35,233,214)(36,234,215)(37,235,216)(38,236,217)(39,237,218)(40,238,219)(71,87,105)(72,88,106)(73,89,107)(74,90,108)(75,81,109)(76,82,110)(77,83,101)(78,84,102)(79,85,103)(80,86,104)(91,121,112)(92,122,113)(93,123,114)(94,124,115)(95,125,116)(96,126,117)(97,127,118)(98,128,119)(99,129,120)(100,130,111)(131,145,162)(132,146,163)(133,147,164)(134,148,165)(135,149,166)(136,150,167)(137,141,168)(138,142,169)(139,143,170)(140,144,161)(151,189,180)(152,190,171)(153,181,172)(154,182,173)(155,183,174)(156,184,175)(157,185,176)(158,186,177)(159,187,178)(160,188,179)(191,205,222)(192,206,223)(193,207,224)(194,208,225)(195,209,226)(196,210,227)(197,201,228)(198,202,229)(199,203,230)(200,204,221), (11,67)(12,68)(13,69)(14,70)(15,61)(16,62)(17,63)(18,64)(19,65)(20,66)(21,49)(22,50)(23,41)(24,42)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,220)(32,211)(33,212)(34,213)(35,214)(36,215)(37,216)(38,217)(39,218)(40,219)(81,109)(82,110)(83,101)(84,102)(85,103)(86,104)(87,105)(88,106)(89,107)(90,108)(91,121)(92,122)(93,123)(94,124)(95,125)(96,126)(97,127)(98,128)(99,129)(100,130)(141,168)(142,169)(143,170)(144,161)(145,162)(146,163)(147,164)(148,165)(149,166)(150,167)(151,189)(152,190)(153,181)(154,182)(155,183)(156,184)(157,185)(158,186)(159,187)(160,188)(201,228)(202,229)(203,230)(204,221)(205,222)(206,223)(207,224)(208,225)(209,226)(210,227), (1,111,51,71)(2,112,52,72)(3,113,53,73)(4,114,54,74)(5,115,55,75)(6,116,56,76)(7,117,57,77)(8,118,58,78)(9,119,59,79)(10,120,60,80)(11,87,27,100)(12,88,28,91)(13,89,29,92)(14,90,30,93)(15,81,21,94)(16,82,22,95)(17,83,23,96)(18,84,24,97)(19,85,25,98)(20,86,26,99)(31,165,225,188)(32,166,226,189)(33,167,227,190)(34,168,228,181)(35,169,229,182)(36,170,230,183)(37,161,221,184)(38,162,222,185)(39,163,223,186)(40,164,224,187)(41,126,63,101)(42,127,64,102)(43,128,65,103)(44,129,66,104)(45,130,67,105)(46,121,68,106)(47,122,69,107)(48,123,70,108)(49,124,61,109)(50,125,62,110)(131,191,176,236)(132,192,177,237)(133,193,178,238)(134,194,179,239)(135,195,180,240)(136,196,171,231)(137,197,172,232)(138,198,173,233)(139,199,174,234)(140,200,175,235)(141,201,153,213)(142,202,154,214)(143,203,155,215)(144,204,156,216)(145,205,157,217)(146,206,158,218)(147,207,159,219)(148,208,160,220)(149,209,151,211)(150,210,152,212), (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), (1,139,6,134)(2,138,7,133)(3,137,8,132)(4,136,9,131)(5,135,10,140)(11,155,16,160)(12,154,17,159)(13,153,18,158)(14,152,19,157)(15,151,20,156)(21,149,26,144)(22,148,27,143)(23,147,28,142)(24,146,29,141)(25,145,30,150)(31,130,36,125)(32,129,37,124)(33,128,38,123)(34,127,39,122)(35,126,40,121)(41,164,46,169)(42,163,47,168)(43,162,48,167)(44,161,49,166)(45,170,50,165)(51,174,56,179)(52,173,57,178)(53,172,58,177)(54,171,59,176)(55,180,60,175)(61,189,66,184)(62,188,67,183)(63,187,68,182)(64,186,69,181)(65,185,70,190)(71,199,76,194)(72,198,77,193)(73,197,78,192)(74,196,79,191)(75,195,80,200)(81,209,86,204)(82,208,87,203)(83,207,88,202)(84,206,89,201)(85,205,90,210)(91,214,96,219)(92,213,97,218)(93,212,98,217)(94,211,99,216)(95,220,100,215)(101,224,106,229)(102,223,107,228)(103,222,108,227)(104,221,109,226)(105,230,110,225)(111,234,116,239)(112,233,117,238)(113,232,118,237)(114,231,119,236)(115,240,120,235) );
G=PermutationGroup([[(1,27,45),(2,28,46),(3,29,47),(4,30,48),(5,21,49),(6,22,50),(7,23,41),(8,24,42),(9,25,43),(10,26,44),(11,67,51),(12,68,52),(13,69,53),(14,70,54),(15,61,55),(16,62,56),(17,63,57),(18,64,58),(19,65,59),(20,66,60),(31,239,220),(32,240,211),(33,231,212),(34,232,213),(35,233,214),(36,234,215),(37,235,216),(38,236,217),(39,237,218),(40,238,219),(71,87,105),(72,88,106),(73,89,107),(74,90,108),(75,81,109),(76,82,110),(77,83,101),(78,84,102),(79,85,103),(80,86,104),(91,121,112),(92,122,113),(93,123,114),(94,124,115),(95,125,116),(96,126,117),(97,127,118),(98,128,119),(99,129,120),(100,130,111),(131,145,162),(132,146,163),(133,147,164),(134,148,165),(135,149,166),(136,150,167),(137,141,168),(138,142,169),(139,143,170),(140,144,161),(151,189,180),(152,190,171),(153,181,172),(154,182,173),(155,183,174),(156,184,175),(157,185,176),(158,186,177),(159,187,178),(160,188,179),(191,205,222),(192,206,223),(193,207,224),(194,208,225),(195,209,226),(196,210,227),(197,201,228),(198,202,229),(199,203,230),(200,204,221)], [(11,67),(12,68),(13,69),(14,70),(15,61),(16,62),(17,63),(18,64),(19,65),(20,66),(21,49),(22,50),(23,41),(24,42),(25,43),(26,44),(27,45),(28,46),(29,47),(30,48),(31,220),(32,211),(33,212),(34,213),(35,214),(36,215),(37,216),(38,217),(39,218),(40,219),(81,109),(82,110),(83,101),(84,102),(85,103),(86,104),(87,105),(88,106),(89,107),(90,108),(91,121),(92,122),(93,123),(94,124),(95,125),(96,126),(97,127),(98,128),(99,129),(100,130),(141,168),(142,169),(143,170),(144,161),(145,162),(146,163),(147,164),(148,165),(149,166),(150,167),(151,189),(152,190),(153,181),(154,182),(155,183),(156,184),(157,185),(158,186),(159,187),(160,188),(201,228),(202,229),(203,230),(204,221),(205,222),(206,223),(207,224),(208,225),(209,226),(210,227)], [(1,111,51,71),(2,112,52,72),(3,113,53,73),(4,114,54,74),(5,115,55,75),(6,116,56,76),(7,117,57,77),(8,118,58,78),(9,119,59,79),(10,120,60,80),(11,87,27,100),(12,88,28,91),(13,89,29,92),(14,90,30,93),(15,81,21,94),(16,82,22,95),(17,83,23,96),(18,84,24,97),(19,85,25,98),(20,86,26,99),(31,165,225,188),(32,166,226,189),(33,167,227,190),(34,168,228,181),(35,169,229,182),(36,170,230,183),(37,161,221,184),(38,162,222,185),(39,163,223,186),(40,164,224,187),(41,126,63,101),(42,127,64,102),(43,128,65,103),(44,129,66,104),(45,130,67,105),(46,121,68,106),(47,122,69,107),(48,123,70,108),(49,124,61,109),(50,125,62,110),(131,191,176,236),(132,192,177,237),(133,193,178,238),(134,194,179,239),(135,195,180,240),(136,196,171,231),(137,197,172,232),(138,198,173,233),(139,199,174,234),(140,200,175,235),(141,201,153,213),(142,202,154,214),(143,203,155,215),(144,204,156,216),(145,205,157,217),(146,206,158,218),(147,207,159,219),(148,208,160,220),(149,209,151,211),(150,210,152,212)], [(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)], [(1,139,6,134),(2,138,7,133),(3,137,8,132),(4,136,9,131),(5,135,10,140),(11,155,16,160),(12,154,17,159),(13,153,18,158),(14,152,19,157),(15,151,20,156),(21,149,26,144),(22,148,27,143),(23,147,28,142),(24,146,29,141),(25,145,30,150),(31,130,36,125),(32,129,37,124),(33,128,38,123),(34,127,39,122),(35,126,40,121),(41,164,46,169),(42,163,47,168),(43,162,48,167),(44,161,49,166),(45,170,50,165),(51,174,56,179),(52,173,57,178),(53,172,58,177),(54,171,59,176),(55,180,60,175),(61,189,66,184),(62,188,67,183),(63,187,68,182),(64,186,69,181),(65,185,70,190),(71,199,76,194),(72,198,77,193),(73,197,78,192),(74,196,79,191),(75,195,80,200),(81,209,86,204),(82,208,87,203),(83,207,88,202),(84,206,89,201),(85,205,90,210),(91,214,96,219),(92,213,97,218),(93,212,98,217),(94,211,99,216),(95,220,100,215),(101,224,106,229),(102,223,107,228),(103,222,108,227),(104,221,109,226),(105,230,110,225),(111,234,116,239),(112,233,117,238),(113,232,118,237),(114,231,119,236),(115,240,120,235)]])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | ··· | 20H | 20I | ··· | 20P | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 2 | 2 | 2 | 6 | 6 | 10 | 10 | 10 | 10 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | - | + | + | + | - | + | + | - | + | - | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | Q8 | D5 | D6 | D6 | Dic5 | D10 | D10 | D10 | C4×S3 | Dic10 | D20 | S3×D4 | S3×Q8 | S3×D5 | S3×Dic5 | C2×S3×D5 | S3×Dic10 | S3×D20 |
| kernel | S3×C4⋊Dic5 | C6.Dic10 | C3×C4⋊Dic5 | C60⋊5C4 | C2×S3×Dic5 | S3×C2×C20 | S3×C20 | C4⋊Dic5 | S3×C10 | S3×C10 | S3×C2×C4 | C2×Dic5 | C2×C20 | C4×S3 | C2×Dic3 | C2×C12 | C22×S3 | C20 | D6 | D6 | C10 | C10 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 1 | 2 | 2 | 2 | 2 | 1 | 8 | 2 | 2 | 2 | 4 | 8 | 8 | 1 | 1 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of S3×C4⋊Dic5 ►in GL4(𝔽61) generated by
| 60 | 60 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 60 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 29 | 7 |
| 0 | 0 | 54 | 32 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 0 | 1 | 17 |
| 11 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 2 | 32 |
| 0 | 0 | 59 | 59 |
G:=sub<GL(4,GF(61))| [60,1,0,0,60,0,0,0,0,0,1,0,0,0,0,1],[1,60,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,29,54,0,0,7,32],[60,0,0,0,0,60,0,0,0,0,0,1,0,0,60,17],[11,0,0,0,0,11,0,0,0,0,2,59,0,0,32,59] >;
S3×C4⋊Dic5 in GAP, Magma, Sage, TeX
S_3\times C_4\rtimes {\rm Dic}_5 % in TeX
G:=Group("S3xC4:Dic5"); // GroupNames label
G:=SmallGroup(480,502);
// by ID
G=gap.SmallGroup(480,502);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,422,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^4=d^10=1,e^2=d^5,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations