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G = (S3×Dic5)⋊C4order 480 = 25·3·5

3rd semidirect product of S3×Dic5 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D6⋊C4.6D5, D6.8(C4×D5), (S3×Dic5)⋊3C4, (C2×C20).193D6, (C4×Dic15)⋊11C2, (C2×C12).192D10, C10.D412S3, D6⋊Dic5.10C2, C30.52(C22×C4), (C2×C30).90C23, C30.Q812C2, Dic5.23(C4×S3), C1515(C42⋊C2), (Dic3×Dic5)⋊15C2, C30.113(C4○D4), C6.69(D42D5), C2.3(D12⋊D5), (C2×C60).165C22, (C2×Dic5).102D6, (C2×Dic3).93D10, Dic15.39(C2×C4), (C22×S3).33D10, C10.69(D42S3), C10.30(Q83S3), C2.3(C30.C23), C32(C23.11D10), (C6×Dic5).53C22, (C10×Dic3).52C22, (C2×Dic15).202C22, C6.20(C2×C4×D5), C2.22(C4×S3×D5), C10.52(S3×C2×C4), C54(C4⋊C47S3), (C5×D6⋊C4).6C2, (C2×S3×Dic5).3C2, C22.45(C2×S3×D5), (C2×C4).178(S3×D5), (S3×C10).23(C2×C4), (S3×C2×C10).11C22, (C3×C10.D4)⋊12C2, (C3×Dic5).11(C2×C4), (C2×C6).102(C22×D5), (C2×C10).102(C22×S3), SmallGroup(480,476)

Series: Derived Chief Lower central Upper central

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

Generators and relations for (S3×Dic5)⋊C4
 G = < a,b,c,d,e | a3=b2=c10=e4=1, d2=c5, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=bc5, dcd-1=c-1, ce=ec, ede-1=c5d >

Subgroups: 620 in 152 conjugacy classes, 58 normal (44 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×8], C22, C22 [×4], C5, S3 [×2], C6 [×3], C2×C4, C2×C4 [×9], C23, C10 [×3], C10 [×2], Dic3 [×4], C12 [×4], D6 [×2], D6 [×2], C2×C6, C15, C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, Dic5 [×2], Dic5 [×4], C20 [×2], C2×C10, C2×C10 [×4], C4×S3 [×4], C2×Dic3, C2×Dic3 [×2], C2×C12, C2×C12 [×2], C22×S3, C5×S3 [×2], C30 [×3], C42⋊C2, C2×Dic5 [×2], C2×Dic5 [×6], C2×C20, C2×C20, C22×C10, C4×Dic3 [×2], C4⋊Dic3, D6⋊C4, D6⋊C4, C3×C4⋊C4, S3×C2×C4, C5×Dic3, C3×Dic5 [×2], C3×Dic5, Dic15 [×2], Dic15, C60, S3×C10 [×2], S3×C10 [×2], C2×C30, C4×Dic5 [×2], C10.D4, C10.D4, C23.D5, C5×C22⋊C4, C22×Dic5, C4⋊C47S3, S3×Dic5 [×4], C6×Dic5 [×2], C10×Dic3, C2×Dic15 [×2], C2×C60, S3×C2×C10, C23.11D10, Dic3×Dic5, D6⋊Dic5, C30.Q8, C3×C10.D4, C5×D6⋊C4, C4×Dic15, C2×S3×Dic5, (S3×Dic5)⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D5, D6 [×3], C22×C4, C4○D4 [×2], D10 [×3], C4×S3 [×2], C22×S3, C42⋊C2, C4×D5 [×2], C22×D5, S3×C2×C4, D42S3, Q83S3, S3×D5, C2×C4×D5, D42D5 [×2], C4⋊C47S3, C2×S3×D5, C23.11D10, D12⋊D5, C4×S3×D5, C30.C23, (S3×Dic5)⋊C4

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

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

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

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

66 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H4I4J4K4L4M4N5A5B6A6B6C10A···10F10G10H10I10J12A12B12C12D12E12F15A15B20A20B20C20D20E20F20G20H30A···30F60A···60H
order1222223444444444444445566610···10101010101212121212121515202020202020202030···3060···60
size1111662226610101010151515153030222222···2121212124420202020444444121212124···44···4

66 irreducible representations

dim111111111222222222244444444
type+++++++++++++++-++-+-
imageC1C2C2C2C2C2C2C2C4S3D5D6D6C4○D4D10D10D10C4×S3C4×D5D42S3Q83S3S3×D5D42D5C2×S3×D5D12⋊D5C4×S3×D5C30.C23
kernel(S3×Dic5)⋊C4Dic3×Dic5D6⋊Dic5C30.Q8C3×C10.D4C5×D6⋊C4C4×Dic15C2×S3×Dic5S3×Dic5C10.D4D6⋊C4C2×Dic5C2×C20C30C2×Dic3C2×C12C22×S3Dic5D6C10C10C2×C4C6C22C2C2C2
# reps111111118122142224811242444

Matrix representation of (S3×Dic5)⋊C4 in GL6(𝔽61)

100000
010000
001000
000100
0000601
0000600
,
6000000
1510000
0060000
0006000
0000060
0000600
,
6000000
0600000
00446000
00456000
000010
000001
,
5000000
43110000
0060000
0016100
0000600
0000060
,
1530000
0600000
0050000
0005000
0000110
0000011

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,60,0,0,0,0,1,0],[60,15,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,60,0],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,44,45,0,0,0,0,60,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[50,43,0,0,0,0,0,11,0,0,0,0,0,0,60,16,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,53,60,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,11,0,0,0,0,0,0,11] >;

(S3×Dic5)⋊C4 in GAP, Magma, Sage, TeX

(S_3\times {\rm Dic}_5)\rtimes C_4
% in TeX

G:=Group("(S3xDic5):C4");
// GroupNames label

G:=SmallGroup(480,476);
// by ID

G=gap.SmallGroup(480,476);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,422,219,58,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^10=e^4=1,d^2=c^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,e*b*e^-1=b*c^5,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=c^5*d>;
// generators/relations

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