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G = D10⋊C4⋊S3order 480 = 25·3·5

2nd semidirect product of D10⋊C4 and S3 acting via S3/C3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D10⋊C42S3, D6⋊Dic517C2, (S3×C10).30D4, (C2×C20).268D6, C10.135(S3×D4), C30.148(C2×D4), C56(C23.9D6), C6.39(C4○D20), C30.86(C4○D4), (C2×C12).199D10, D6.12(C5⋊D4), C30.Q826C2, C30.4Q833C2, (C2×Dic5).44D6, (C22×D5).13D6, C10.42(C4○D12), D10⋊Dic315C2, (C2×C60).412C22, (C2×C30).142C23, (C22×S3).73D10, C10.32(D42S3), C2.17(D205S3), (C2×Dic3).154D10, C1514(C22.D4), C32(C23.23D10), (C6×Dic5).84C22, C2.28(D6.D10), (C2×Dic15).109C22, (C10×Dic3).188C22, (S3×C2×C4)⋊12D5, (S3×C2×C20)⋊20C2, (C2×C4).80(S3×D5), C6.36(C2×C5⋊D4), C2.15(S3×C5⋊D4), (C2×C15⋊D4).6C2, (D5×C2×C6).27C22, C22.194(C2×S3×D5), (S3×C2×C10).88C22, (C3×D10⋊C4)⋊33C2, (C2×C6).154(C22×D5), (C2×C10).154(C22×S3), SmallGroup(480,528)

Series: Derived Chief Lower central Upper central

C1C2×C30 — D10⋊C4⋊S3
C1C5C15C30C2×C30D5×C2×C6C2×C15⋊D4 — D10⋊C4⋊S3
C15C2×C30 — D10⋊C4⋊S3
C1C22C2×C4

Generators and relations for D10⋊C4⋊S3
 G = < a,b,c,d,e | a10=b2=c4=d3=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, cbc-1=a5b, bd=db, ebe=bc2, cd=dc, ce=ec, ede=d-1 >

Subgroups: 748 in 156 conjugacy classes, 48 normal (44 characteristic)
C1, C2 [×3], C2 [×3], C3, C4 [×5], C22, C22 [×7], C5, S3 [×2], C6 [×3], C6, C2×C4, C2×C4 [×6], D4 [×2], C23 [×2], D5, C10 [×3], C10 [×2], Dic3 [×3], C12 [×2], D6 [×2], D6 [×2], C2×C6, C2×C6 [×3], C15, C22⋊C4 [×3], C4⋊C4 [×2], C22×C4, C2×D4, Dic5 [×3], C20 [×2], D10 [×3], C2×C10, C2×C10 [×4], C4×S3 [×2], C2×Dic3, C2×Dic3 [×2], C3⋊D4 [×2], C2×C12, C2×C12, C22×S3, C22×C6, C5×S3 [×2], C3×D5, C30 [×3], C22.D4, C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×2], C2×C20, C2×C20 [×3], C22×D5, C22×C10, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, S3×C2×C4, C2×C3⋊D4, C5×Dic3, C3×Dic5, Dic15 [×2], C60, C6×D5 [×3], S3×C10 [×2], S3×C10 [×2], C2×C30, C10.D4 [×2], D10⋊C4, D10⋊C4, C23.D5, C2×C5⋊D4, C22×C20, C23.9D6, C15⋊D4 [×2], C6×Dic5, S3×C20 [×2], C10×Dic3, C2×Dic15 [×2], C2×C60, D5×C2×C6, S3×C2×C10, C23.23D10, D10⋊Dic3, D6⋊Dic5, C30.Q8, C3×D10⋊C4, C30.4Q8, C2×C15⋊D4, S3×C2×C20, D10⋊C4⋊S3
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, C4○D4 [×2], D10 [×3], C22×S3, C22.D4, C5⋊D4 [×2], C22×D5, C4○D12, S3×D4, D42S3, S3×D5, C4○D20 [×2], C2×C5⋊D4, C23.9D6, C2×S3×D5, C23.23D10, D205S3, D6.D10, S3×C5⋊D4, D10⋊C4⋊S3

Smallest permutation representation of D10⋊C4⋊S3
On 240 points
Generators in S240
(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)
(1 172)(2 171)(3 180)(4 179)(5 178)(6 177)(7 176)(8 175)(9 174)(10 173)(11 100)(12 99)(13 98)(14 97)(15 96)(16 95)(17 94)(18 93)(19 92)(20 91)(21 167)(22 166)(23 165)(24 164)(25 163)(26 162)(27 161)(28 170)(29 169)(30 168)(31 183)(32 182)(33 181)(34 190)(35 189)(36 188)(37 187)(38 186)(39 185)(40 184)(41 133)(42 132)(43 131)(44 140)(45 139)(46 138)(47 137)(48 136)(49 135)(50 134)(51 148)(52 147)(53 146)(54 145)(55 144)(56 143)(57 142)(58 141)(59 150)(60 149)(61 153)(62 152)(63 151)(64 160)(65 159)(66 158)(67 157)(68 156)(69 155)(70 154)(71 223)(72 222)(73 221)(74 230)(75 229)(76 228)(77 227)(78 226)(79 225)(80 224)(81 238)(82 237)(83 236)(84 235)(85 234)(86 233)(87 232)(88 231)(89 240)(90 239)(101 193)(102 192)(103 191)(104 200)(105 199)(106 198)(107 197)(108 196)(109 195)(110 194)(111 208)(112 207)(113 206)(114 205)(115 204)(116 203)(117 202)(118 201)(119 210)(120 209)(121 212)(122 211)(123 220)(124 219)(125 218)(126 217)(127 216)(128 215)(129 214)(130 213)
(1 117 57 87)(2 118 58 88)(3 119 59 89)(4 120 60 90)(5 111 51 81)(6 112 52 82)(7 113 53 83)(8 114 54 84)(9 115 55 85)(10 116 56 86)(11 189 214 159)(12 190 215 160)(13 181 216 151)(14 182 217 152)(15 183 218 153)(16 184 219 154)(17 185 220 155)(18 186 211 156)(19 187 212 157)(20 188 213 158)(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)(31 130 61 91)(32 121 62 92)(33 122 63 93)(34 123 64 94)(35 124 65 95)(36 125 66 96)(37 126 67 97)(38 127 68 98)(39 128 69 99)(40 129 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)
(1 29 40)(2 30 31)(3 21 32)(4 22 33)(5 23 34)(6 24 35)(7 25 36)(8 26 37)(9 27 38)(10 28 39)(11 232 229)(12 233 230)(13 234 221)(14 235 222)(15 236 223)(16 237 224)(17 238 225)(18 239 226)(19 240 227)(20 231 228)(41 66 53)(42 67 54)(43 68 55)(44 69 56)(45 70 57)(46 61 58)(47 62 59)(48 63 60)(49 64 51)(50 65 52)(71 96 83)(72 97 84)(73 98 85)(74 99 86)(75 100 87)(76 91 88)(77 92 89)(78 93 90)(79 94 81)(80 95 82)(101 125 113)(102 126 114)(103 127 115)(104 128 116)(105 129 117)(106 130 118)(107 121 119)(108 122 120)(109 123 111)(110 124 112)(131 156 144)(132 157 145)(133 158 146)(134 159 147)(135 160 148)(136 151 149)(137 152 150)(138 153 141)(139 154 142)(140 155 143)(161 186 174)(162 187 175)(163 188 176)(164 189 177)(165 190 178)(166 181 179)(167 182 180)(168 183 171)(169 184 172)(170 185 173)(191 216 204)(192 217 205)(193 218 206)(194 219 207)(195 220 208)(196 211 209)(197 212 210)(198 213 201)(199 214 202)(200 215 203)
(11 199)(12 200)(13 191)(14 192)(15 193)(16 194)(17 195)(18 196)(19 197)(20 198)(21 32)(22 33)(23 34)(24 35)(25 36)(26 37)(27 38)(28 39)(29 40)(30 31)(41 66)(42 67)(43 68)(44 69)(45 70)(46 61)(47 62)(48 63)(49 64)(50 65)(71 96)(72 97)(73 98)(74 99)(75 100)(76 91)(77 92)(78 93)(79 94)(80 95)(101 125)(102 126)(103 127)(104 128)(105 129)(106 130)(107 121)(108 122)(109 123)(110 124)(131 186)(132 187)(133 188)(134 189)(135 190)(136 181)(137 182)(138 183)(139 184)(140 185)(141 171)(142 172)(143 173)(144 174)(145 175)(146 176)(147 177)(148 178)(149 179)(150 180)(151 166)(152 167)(153 168)(154 169)(155 170)(156 161)(157 162)(158 163)(159 164)(160 165)(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)

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

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

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

72 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G5A5B6A6B6C6D6E10A···10F10G···10N12A12B12C12D15A15B20A···20H20I···20P30A···30F60A···60H
order122222234444444556666610···1010···1012121212151520···2020···2030···3060···60
size11116620222662060602222220202···26···6442020442···26···64···44···4

72 irreducible representations

dim1111111122222222222224444444
type++++++++++++++++++-++-
imageC1C2C2C2C2C2C2C2S3D4D5D6D6D6C4○D4D10D10D10C5⋊D4C4○D12C4○D20S3×D4D42S3S3×D5C2×S3×D5D205S3D6.D10S3×C5⋊D4
kernelD10⋊C4⋊S3D10⋊Dic3D6⋊Dic5C30.Q8C3×D10⋊C4C30.4Q8C2×C15⋊D4S3×C2×C20D10⋊C4S3×C10S3×C2×C4C2×Dic5C2×C20C22×D5C30C2×Dic3C2×C12C22×S3D6C10C6C10C10C2×C4C22C2C2C2
# reps11111111122111422284161122444

Matrix representation of D10⋊C4⋊S3 in GL4(𝔽61) generated by

52000
02700
0010
0001
,
03400
9000
00943
001852
,
50000
01100
00500
00050
,
1000
0100
00060
00160
,
1000
06000
00160
00060
G:=sub<GL(4,GF(61))| [52,0,0,0,0,27,0,0,0,0,1,0,0,0,0,1],[0,9,0,0,34,0,0,0,0,0,9,18,0,0,43,52],[50,0,0,0,0,11,0,0,0,0,50,0,0,0,0,50],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,60,60],[1,0,0,0,0,60,0,0,0,0,1,0,0,0,60,60] >;

D10⋊C4⋊S3 in GAP, Magma, Sage, TeX

D_{10}\rtimes C_4\rtimes S_3
% in TeX

G:=Group("D10:C4:S3");
// GroupNames label

G:=SmallGroup(480,528);
// by ID

G=gap.SmallGroup(480,528);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,422,58,1356,18822]);
// Polycyclic

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

׿
×
𝔽