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G = S3×C4⋊Dic5order 480 = 25·3·5

Direct product of S3 and C4⋊Dic5

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: S3×C4⋊Dic5, D6.13D20, D6.2Dic10, (S3×C20)⋊6C4, C2017(C4×S3), C6015(C2×C4), C43(S3×Dic5), C2.3(S3×D20), (C4×S3)⋊2Dic5, C605C426C2, C121(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), Dic33(C2×Dic5), (C2×C12).125D10, C6.Dic1020C2, 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, C56(S3×C4⋊C4), C159(C2×C4⋊C4), C31(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

C1C30 — S3×C4⋊Dic5
C1C5C15C30C2×C30C6×Dic5C2×S3×Dic5 — S3×C4⋊Dic5
C15C30 — S3×C4⋊Dic5
C1C22C2×C4

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 [×3], C2 [×4], C3, C4 [×2], C4 [×6], C22, C22 [×6], C5, S3 [×4], C6 [×3], C2×C4, C2×C4 [×13], C23, C10 [×3], C10 [×4], Dic3 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×6], C2×C6, C15, C4⋊C4 [×4], C22×C4 [×3], Dic5 [×4], C20 [×2], C20 [×2], C2×C10, C2×C10 [×6], C4×S3 [×4], C4×S3 [×4], C2×Dic3, C2×Dic3 [×2], C2×C12, C2×C12 [×2], C22×S3, C5×S3 [×4], C30 [×3], C2×C4⋊C4, C2×Dic5 [×2], C2×Dic5 [×6], C2×C20, C2×C20 [×5], C22×C10, Dic3⋊C4 [×2], C4⋊Dic3, C3×C4⋊C4, S3×C2×C4, S3×C2×C4 [×2], C5×Dic3 [×2], C3×Dic5 [×2], Dic15 [×2], C60 [×2], S3×C10 [×6], C2×C30, C4⋊Dic5, C4⋊Dic5 [×3], C22×Dic5 [×2], C22×C20, S3×C4⋊C4, S3×Dic5 [×4], C6×Dic5 [×2], S3×C20 [×4], C10×Dic3, C2×Dic15 [×2], C2×C60, S3×C2×C10, C2×C4⋊Dic5, C6.Dic10 [×2], C3×C4⋊Dic5, C605C4, C2×S3×Dic5 [×2], S3×C2×C20, S3×C4⋊Dic5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, D6 [×3], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, Dic5 [×4], D10 [×3], C4×S3 [×2], C22×S3, C2×C4⋊C4, Dic10 [×2], D20 [×2], C2×Dic5 [×6], C22×D5, S3×C2×C4, S3×D4, S3×Q8, S3×D5, C4⋊Dic5 [×4], C2×Dic10, C2×D20, C22×Dic5, S3×C4⋊C4, S3×Dic5 [×2], C2×S3×D5, C2×C4⋊Dic5, S3×Dic10, S3×D20, C2×S3×Dic5, S3×C4⋊Dic5

Smallest permutation representation of S3×C4⋊Dic5
On 240 points
Generators in S240
(1 27 41)(2 28 42)(3 29 43)(4 30 44)(5 21 45)(6 22 46)(7 23 47)(8 24 48)(9 25 49)(10 26 50)(11 61 55)(12 62 56)(13 63 57)(14 64 58)(15 65 59)(16 66 60)(17 67 51)(18 68 52)(19 69 53)(20 70 54)(31 239 214)(32 240 215)(33 231 216)(34 232 217)(35 233 218)(36 234 219)(37 235 220)(38 236 211)(39 237 212)(40 238 213)(71 81 105)(72 82 106)(73 83 107)(74 84 108)(75 85 109)(76 86 110)(77 87 101)(78 88 102)(79 89 103)(80 90 104)(91 125 116)(92 126 117)(93 127 118)(94 128 119)(95 129 120)(96 130 111)(97 121 112)(98 122 113)(99 123 114)(100 124 115)(131 141 162)(132 142 163)(133 143 164)(134 144 165)(135 145 166)(136 146 167)(137 147 168)(138 148 169)(139 149 170)(140 150 161)(151 185 176)(152 186 177)(153 187 178)(154 188 179)(155 189 180)(156 190 171)(157 181 172)(158 182 173)(159 183 174)(160 184 175)(191 201 222)(192 202 223)(193 203 224)(194 204 225)(195 205 226)(196 206 227)(197 207 228)(198 208 229)(199 209 230)(200 210 221)
(11 61)(12 62)(13 63)(14 64)(15 65)(16 66)(17 67)(18 68)(19 69)(20 70)(21 45)(22 46)(23 47)(24 48)(25 49)(26 50)(27 41)(28 42)(29 43)(30 44)(31 214)(32 215)(33 216)(34 217)(35 218)(36 219)(37 220)(38 211)(39 212)(40 213)(81 105)(82 106)(83 107)(84 108)(85 109)(86 110)(87 101)(88 102)(89 103)(90 104)(91 125)(92 126)(93 127)(94 128)(95 129)(96 130)(97 121)(98 122)(99 123)(100 124)(141 162)(142 163)(143 164)(144 165)(145 166)(146 167)(147 168)(148 169)(149 170)(150 161)(151 185)(152 186)(153 187)(154 188)(155 189)(156 190)(157 181)(158 182)(159 183)(160 184)(201 222)(202 223)(203 224)(204 225)(205 226)(206 227)(207 228)(208 229)(209 230)(210 221)
(1 111 51 77)(2 112 52 78)(3 113 53 79)(4 114 54 80)(5 115 55 71)(6 116 56 72)(7 117 57 73)(8 118 58 74)(9 119 59 75)(10 120 60 76)(11 81 21 100)(12 82 22 91)(13 83 23 92)(14 84 24 93)(15 85 25 94)(16 86 26 95)(17 87 27 96)(18 88 28 97)(19 89 29 98)(20 90 30 99)(31 169 229 188)(32 170 230 189)(33 161 221 190)(34 162 222 181)(35 163 223 182)(36 164 224 183)(37 165 225 184)(38 166 226 185)(39 167 227 186)(40 168 228 187)(41 130 67 101)(42 121 68 102)(43 122 69 103)(44 123 70 104)(45 124 61 105)(46 125 62 106)(47 126 63 107)(48 127 64 108)(49 128 65 109)(50 129 66 110)(131 191 172 232)(132 192 173 233)(133 193 174 234)(134 194 175 235)(135 195 176 236)(136 196 177 237)(137 197 178 238)(138 198 179 239)(139 199 180 240)(140 200 171 231)(141 201 157 217)(142 202 158 218)(143 203 159 219)(144 204 160 220)(145 205 151 211)(146 206 152 212)(147 207 153 213)(148 208 154 214)(149 209 155 215)(150 210 156 216)
(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)
(1 133 6 138)(2 132 7 137)(3 131 8 136)(4 140 9 135)(5 139 10 134)(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)| (1,27,41)(2,28,42)(3,29,43)(4,30,44)(5,21,45)(6,22,46)(7,23,47)(8,24,48)(9,25,49)(10,26,50)(11,61,55)(12,62,56)(13,63,57)(14,64,58)(15,65,59)(16,66,60)(17,67,51)(18,68,52)(19,69,53)(20,70,54)(31,239,214)(32,240,215)(33,231,216)(34,232,217)(35,233,218)(36,234,219)(37,235,220)(38,236,211)(39,237,212)(40,238,213)(71,81,105)(72,82,106)(73,83,107)(74,84,108)(75,85,109)(76,86,110)(77,87,101)(78,88,102)(79,89,103)(80,90,104)(91,125,116)(92,126,117)(93,127,118)(94,128,119)(95,129,120)(96,130,111)(97,121,112)(98,122,113)(99,123,114)(100,124,115)(131,141,162)(132,142,163)(133,143,164)(134,144,165)(135,145,166)(136,146,167)(137,147,168)(138,148,169)(139,149,170)(140,150,161)(151,185,176)(152,186,177)(153,187,178)(154,188,179)(155,189,180)(156,190,171)(157,181,172)(158,182,173)(159,183,174)(160,184,175)(191,201,222)(192,202,223)(193,203,224)(194,204,225)(195,205,226)(196,206,227)(197,207,228)(198,208,229)(199,209,230)(200,210,221), (11,61)(12,62)(13,63)(14,64)(15,65)(16,66)(17,67)(18,68)(19,69)(20,70)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50)(27,41)(28,42)(29,43)(30,44)(31,214)(32,215)(33,216)(34,217)(35,218)(36,219)(37,220)(38,211)(39,212)(40,213)(81,105)(82,106)(83,107)(84,108)(85,109)(86,110)(87,101)(88,102)(89,103)(90,104)(91,125)(92,126)(93,127)(94,128)(95,129)(96,130)(97,121)(98,122)(99,123)(100,124)(141,162)(142,163)(143,164)(144,165)(145,166)(146,167)(147,168)(148,169)(149,170)(150,161)(151,185)(152,186)(153,187)(154,188)(155,189)(156,190)(157,181)(158,182)(159,183)(160,184)(201,222)(202,223)(203,224)(204,225)(205,226)(206,227)(207,228)(208,229)(209,230)(210,221), (1,111,51,77)(2,112,52,78)(3,113,53,79)(4,114,54,80)(5,115,55,71)(6,116,56,72)(7,117,57,73)(8,118,58,74)(9,119,59,75)(10,120,60,76)(11,81,21,100)(12,82,22,91)(13,83,23,92)(14,84,24,93)(15,85,25,94)(16,86,26,95)(17,87,27,96)(18,88,28,97)(19,89,29,98)(20,90,30,99)(31,169,229,188)(32,170,230,189)(33,161,221,190)(34,162,222,181)(35,163,223,182)(36,164,224,183)(37,165,225,184)(38,166,226,185)(39,167,227,186)(40,168,228,187)(41,130,67,101)(42,121,68,102)(43,122,69,103)(44,123,70,104)(45,124,61,105)(46,125,62,106)(47,126,63,107)(48,127,64,108)(49,128,65,109)(50,129,66,110)(131,191,172,232)(132,192,173,233)(133,193,174,234)(134,194,175,235)(135,195,176,236)(136,196,177,237)(137,197,178,238)(138,198,179,239)(139,199,180,240)(140,200,171,231)(141,201,157,217)(142,202,158,218)(143,203,159,219)(144,204,160,220)(145,205,151,211)(146,206,152,212)(147,207,153,213)(148,208,154,214)(149,209,155,215)(150,210,156,216), (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), (1,133,6,138)(2,132,7,137)(3,131,8,136)(4,140,9,135)(5,139,10,134)(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( (1,27,41)(2,28,42)(3,29,43)(4,30,44)(5,21,45)(6,22,46)(7,23,47)(8,24,48)(9,25,49)(10,26,50)(11,61,55)(12,62,56)(13,63,57)(14,64,58)(15,65,59)(16,66,60)(17,67,51)(18,68,52)(19,69,53)(20,70,54)(31,239,214)(32,240,215)(33,231,216)(34,232,217)(35,233,218)(36,234,219)(37,235,220)(38,236,211)(39,237,212)(40,238,213)(71,81,105)(72,82,106)(73,83,107)(74,84,108)(75,85,109)(76,86,110)(77,87,101)(78,88,102)(79,89,103)(80,90,104)(91,125,116)(92,126,117)(93,127,118)(94,128,119)(95,129,120)(96,130,111)(97,121,112)(98,122,113)(99,123,114)(100,124,115)(131,141,162)(132,142,163)(133,143,164)(134,144,165)(135,145,166)(136,146,167)(137,147,168)(138,148,169)(139,149,170)(140,150,161)(151,185,176)(152,186,177)(153,187,178)(154,188,179)(155,189,180)(156,190,171)(157,181,172)(158,182,173)(159,183,174)(160,184,175)(191,201,222)(192,202,223)(193,203,224)(194,204,225)(195,205,226)(196,206,227)(197,207,228)(198,208,229)(199,209,230)(200,210,221), (11,61)(12,62)(13,63)(14,64)(15,65)(16,66)(17,67)(18,68)(19,69)(20,70)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50)(27,41)(28,42)(29,43)(30,44)(31,214)(32,215)(33,216)(34,217)(35,218)(36,219)(37,220)(38,211)(39,212)(40,213)(81,105)(82,106)(83,107)(84,108)(85,109)(86,110)(87,101)(88,102)(89,103)(90,104)(91,125)(92,126)(93,127)(94,128)(95,129)(96,130)(97,121)(98,122)(99,123)(100,124)(141,162)(142,163)(143,164)(144,165)(145,166)(146,167)(147,168)(148,169)(149,170)(150,161)(151,185)(152,186)(153,187)(154,188)(155,189)(156,190)(157,181)(158,182)(159,183)(160,184)(201,222)(202,223)(203,224)(204,225)(205,226)(206,227)(207,228)(208,229)(209,230)(210,221), (1,111,51,77)(2,112,52,78)(3,113,53,79)(4,114,54,80)(5,115,55,71)(6,116,56,72)(7,117,57,73)(8,118,58,74)(9,119,59,75)(10,120,60,76)(11,81,21,100)(12,82,22,91)(13,83,23,92)(14,84,24,93)(15,85,25,94)(16,86,26,95)(17,87,27,96)(18,88,28,97)(19,89,29,98)(20,90,30,99)(31,169,229,188)(32,170,230,189)(33,161,221,190)(34,162,222,181)(35,163,223,182)(36,164,224,183)(37,165,225,184)(38,166,226,185)(39,167,227,186)(40,168,228,187)(41,130,67,101)(42,121,68,102)(43,122,69,103)(44,123,70,104)(45,124,61,105)(46,125,62,106)(47,126,63,107)(48,127,64,108)(49,128,65,109)(50,129,66,110)(131,191,172,232)(132,192,173,233)(133,193,174,234)(134,194,175,235)(135,195,176,236)(136,196,177,237)(137,197,178,238)(138,198,179,239)(139,199,180,240)(140,200,171,231)(141,201,157,217)(142,202,158,218)(143,203,159,219)(144,204,160,220)(145,205,151,211)(146,206,152,212)(147,207,153,213)(148,208,154,214)(149,209,155,215)(150,210,156,216), (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), (1,133,6,138)(2,132,7,137)(3,131,8,136)(4,140,9,135)(5,139,10,134)(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([(1,27,41),(2,28,42),(3,29,43),(4,30,44),(5,21,45),(6,22,46),(7,23,47),(8,24,48),(9,25,49),(10,26,50),(11,61,55),(12,62,56),(13,63,57),(14,64,58),(15,65,59),(16,66,60),(17,67,51),(18,68,52),(19,69,53),(20,70,54),(31,239,214),(32,240,215),(33,231,216),(34,232,217),(35,233,218),(36,234,219),(37,235,220),(38,236,211),(39,237,212),(40,238,213),(71,81,105),(72,82,106),(73,83,107),(74,84,108),(75,85,109),(76,86,110),(77,87,101),(78,88,102),(79,89,103),(80,90,104),(91,125,116),(92,126,117),(93,127,118),(94,128,119),(95,129,120),(96,130,111),(97,121,112),(98,122,113),(99,123,114),(100,124,115),(131,141,162),(132,142,163),(133,143,164),(134,144,165),(135,145,166),(136,146,167),(137,147,168),(138,148,169),(139,149,170),(140,150,161),(151,185,176),(152,186,177),(153,187,178),(154,188,179),(155,189,180),(156,190,171),(157,181,172),(158,182,173),(159,183,174),(160,184,175),(191,201,222),(192,202,223),(193,203,224),(194,204,225),(195,205,226),(196,206,227),(197,207,228),(198,208,229),(199,209,230),(200,210,221)], [(11,61),(12,62),(13,63),(14,64),(15,65),(16,66),(17,67),(18,68),(19,69),(20,70),(21,45),(22,46),(23,47),(24,48),(25,49),(26,50),(27,41),(28,42),(29,43),(30,44),(31,214),(32,215),(33,216),(34,217),(35,218),(36,219),(37,220),(38,211),(39,212),(40,213),(81,105),(82,106),(83,107),(84,108),(85,109),(86,110),(87,101),(88,102),(89,103),(90,104),(91,125),(92,126),(93,127),(94,128),(95,129),(96,130),(97,121),(98,122),(99,123),(100,124),(141,162),(142,163),(143,164),(144,165),(145,166),(146,167),(147,168),(148,169),(149,170),(150,161),(151,185),(152,186),(153,187),(154,188),(155,189),(156,190),(157,181),(158,182),(159,183),(160,184),(201,222),(202,223),(203,224),(204,225),(205,226),(206,227),(207,228),(208,229),(209,230),(210,221)], [(1,111,51,77),(2,112,52,78),(3,113,53,79),(4,114,54,80),(5,115,55,71),(6,116,56,72),(7,117,57,73),(8,118,58,74),(9,119,59,75),(10,120,60,76),(11,81,21,100),(12,82,22,91),(13,83,23,92),(14,84,24,93),(15,85,25,94),(16,86,26,95),(17,87,27,96),(18,88,28,97),(19,89,29,98),(20,90,30,99),(31,169,229,188),(32,170,230,189),(33,161,221,190),(34,162,222,181),(35,163,223,182),(36,164,224,183),(37,165,225,184),(38,166,226,185),(39,167,227,186),(40,168,228,187),(41,130,67,101),(42,121,68,102),(43,122,69,103),(44,123,70,104),(45,124,61,105),(46,125,62,106),(47,126,63,107),(48,127,64,108),(49,128,65,109),(50,129,66,110),(131,191,172,232),(132,192,173,233),(133,193,174,234),(134,194,175,235),(135,195,176,236),(136,196,177,237),(137,197,178,238),(138,198,179,239),(139,199,180,240),(140,200,171,231),(141,201,157,217),(142,202,158,218),(143,203,159,219),(144,204,160,220),(145,205,151,211),(146,206,152,212),(147,207,153,213),(148,208,154,214),(149,209,155,215),(150,210,156,216)], [(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)], [(1,133,6,138),(2,132,7,137),(3,131,8,136),(4,140,9,135),(5,139,10,134),(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 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G4H4I4J4K4L5A5B6A6B6C10A···10F10G···10N12A12B12C12D12E12F15A15B20A···20H20I···20P30A···30F60A···60H
order1222222234444444444445566610···1010···10121212121212151520···2020···2030···3060···60
size11113333222661010101030303030222222···26···64420202020442···26···64···44···4

78 irreducible representations

dim111111122222222222224444444
type++++++++-+++-+++-++-+-+-+
imageC1C2C2C2C2C2C4S3D4Q8D5D6D6Dic5D10D10D10C4×S3Dic10D20S3×D4S3×Q8S3×D5S3×Dic5C2×S3×D5S3×Dic10S3×D20
kernelS3×C4⋊Dic5C6.Dic10C3×C4⋊Dic5C605C4C2×S3×Dic5S3×C2×C20S3×C20C4⋊Dic5S3×C10S3×C10S3×C2×C4C2×Dic5C2×C20C4×S3C2×Dic3C2×C12C22×S3C20D6D6C10C10C2×C4C4C22C2C2
# reps121121812222182224881124244

Matrix representation of S3×C4⋊Dic5 in GL4(𝔽61) generated by

606000
1000
0010
0001
,
1000
606000
0010
0001
,
1000
0100
00297
005432
,
60000
06000
00060
00117
,
11000
01100
00232
005959
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

׿
×
𝔽