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## G = C10.D4⋊S3order 480 = 25·3·5

### 3rd semidirect product of C10.D4 and S3 acting via S3/C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — C10.D4⋊S3
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — Dic15⋊5C4 — C10.D4⋊S3
 Lower central C15 — C2×C30 — C10.D4⋊S3
 Upper central C1 — C22 — C2×C4

Generators and relations for C10.D4⋊S3
G = < a,b,c,d,e | a10=b4=d3=e2=1, c2=a5, bab-1=cac-1=eae=a-1, ad=da, cbc-1=b-1, bd=db, ebe=a5b, cd=dc, ece=a5b2c, ede=d-1 >

Subgroups: 652 in 120 conjugacy classes, 44 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, C23, D5, C10, Dic3, C12, D6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, Dic5, C20, D10, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, D15, C30, C422C2, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C5×Dic3, C3×Dic5, Dic15, C60, D30, C2×C30, C10.D4, C10.D4, D10⋊C4, C4×C20, C4⋊C4⋊S3, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, C22×D15, C422D5, D304C4, C30.Q8, Dic155C4, C3×C10.D4, Dic3×C20, D303C4, C10.D4⋊S3
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, D10, C22×S3, C422C2, C22×D5, C4○D12, D42S3, Q83S3, S3×D5, C4○D20, C4⋊C4⋊S3, C2×S3×D5, C422D5, D60⋊C2, D6.D10, Dic5.D6, C10.D4⋊S3

Smallest permutation representation of C10.D4⋊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)
(1 237 25 185)(2 236 26 184)(3 235 27 183)(4 234 28 182)(5 233 29 181)(6 232 30 190)(7 231 21 189)(8 240 22 188)(9 239 23 187)(10 238 24 186)(11 221 60 45)(12 230 51 44)(13 229 52 43)(14 228 53 42)(15 227 54 41)(16 226 55 50)(17 225 56 49)(18 224 57 48)(19 223 58 47)(20 222 59 46)(31 191 89 213)(32 200 90 212)(33 199 81 211)(34 198 82 220)(35 197 83 219)(36 196 84 218)(37 195 85 217)(38 194 86 216)(39 193 87 215)(40 192 88 214)(61 176 113 124)(62 175 114 123)(63 174 115 122)(64 173 116 121)(65 172 117 130)(66 171 118 129)(67 180 119 128)(68 179 120 127)(69 178 111 126)(70 177 112 125)(71 204 93 166)(72 203 94 165)(73 202 95 164)(74 201 96 163)(75 210 97 162)(76 209 98 161)(77 208 99 170)(78 207 100 169)(79 206 91 168)(80 205 92 167)(101 158 149 136)(102 157 150 135)(103 156 141 134)(104 155 142 133)(105 154 143 132)(106 153 144 131)(107 152 145 140)(108 151 146 139)(109 160 147 138)(110 159 148 137)
(1 177 6 172)(2 176 7 171)(3 175 8 180)(4 174 9 179)(5 173 10 178)(11 161 16 166)(12 170 17 165)(13 169 18 164)(14 168 19 163)(15 167 20 162)(21 129 26 124)(22 128 27 123)(23 127 28 122)(24 126 29 121)(25 125 30 130)(31 131 36 136)(32 140 37 135)(33 139 38 134)(34 138 39 133)(35 137 40 132)(41 80 46 75)(42 79 47 74)(43 78 48 73)(44 77 49 72)(45 76 50 71)(51 208 56 203)(52 207 57 202)(53 206 58 201)(54 205 59 210)(55 204 60 209)(61 231 66 236)(62 240 67 235)(63 239 68 234)(64 238 69 233)(65 237 70 232)(81 151 86 156)(82 160 87 155)(83 159 88 154)(84 158 89 153)(85 157 90 152)(91 223 96 228)(92 222 97 227)(93 221 98 226)(94 230 99 225)(95 229 100 224)(101 213 106 218)(102 212 107 217)(103 211 108 216)(104 220 109 215)(105 219 110 214)(111 181 116 186)(112 190 117 185)(113 189 118 184)(114 188 119 183)(115 187 120 182)(141 199 146 194)(142 198 147 193)(143 197 148 192)(144 196 149 191)(145 195 150 200)
(1 59 90)(2 60 81)(3 51 82)(4 52 83)(5 53 84)(6 54 85)(7 55 86)(8 56 87)(9 57 88)(10 58 89)(11 33 26)(12 34 27)(13 35 28)(14 36 29)(15 37 30)(16 38 21)(17 39 22)(18 40 23)(19 31 24)(20 32 25)(41 217 232)(42 218 233)(43 219 234)(44 220 235)(45 211 236)(46 212 237)(47 213 238)(48 214 239)(49 215 240)(50 216 231)(61 76 108)(62 77 109)(63 78 110)(64 79 101)(65 80 102)(66 71 103)(67 72 104)(68 73 105)(69 74 106)(70 75 107)(91 149 116)(92 150 117)(93 141 118)(94 142 119)(95 143 120)(96 144 111)(97 145 112)(98 146 113)(99 147 114)(100 148 115)(121 168 136)(122 169 137)(123 170 138)(124 161 139)(125 162 140)(126 163 131)(127 164 132)(128 165 133)(129 166 134)(130 167 135)(151 176 209)(152 177 210)(153 178 201)(154 179 202)(155 180 203)(156 171 204)(157 172 205)(158 173 206)(159 174 207)(160 175 208)(181 228 196)(182 229 197)(183 230 198)(184 221 199)(185 222 200)(186 223 191)(187 224 192)(188 225 193)(189 226 194)(190 227 195)
(2 10)(3 9)(4 8)(5 7)(11 31)(12 40)(13 39)(14 38)(15 37)(16 36)(17 35)(18 34)(19 33)(20 32)(21 29)(22 28)(23 27)(24 26)(41 212)(42 211)(43 220)(44 219)(45 218)(46 217)(47 216)(48 215)(49 214)(50 213)(51 88)(52 87)(53 86)(54 85)(55 84)(56 83)(57 82)(58 81)(59 90)(60 89)(61 111)(62 120)(63 119)(64 118)(65 117)(66 116)(67 115)(68 114)(69 113)(70 112)(71 149)(72 148)(73 147)(74 146)(75 145)(76 144)(77 143)(78 142)(79 141)(80 150)(91 103)(92 102)(93 101)(94 110)(95 109)(96 108)(97 107)(98 106)(99 105)(100 104)(121 176)(122 175)(123 174)(124 173)(125 172)(126 171)(127 180)(128 179)(129 178)(130 177)(131 204)(132 203)(133 202)(134 201)(135 210)(136 209)(137 208)(138 207)(139 206)(140 205)(151 168)(152 167)(153 166)(154 165)(155 164)(156 163)(157 162)(158 161)(159 170)(160 169)(181 184)(182 183)(185 190)(186 189)(187 188)(191 226)(192 225)(193 224)(194 223)(195 222)(196 221)(197 230)(198 229)(199 228)(200 227)(231 238)(232 237)(233 236)(234 235)(239 240)```

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 5A 5B 6A 6B 6C 10A ··· 10F 12A 12B 12C 12D 12E 12F 15A 15B 20A ··· 20H 20I ··· 20X 30A ··· 30F 60A ··· 60H order 1 2 2 2 2 3 4 4 4 4 4 4 4 4 4 5 5 6 6 6 10 ··· 10 12 12 12 12 12 12 15 15 20 ··· 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 60 2 2 2 6 6 6 6 20 20 60 2 2 2 2 2 2 ··· 2 4 4 20 20 20 20 4 4 2 ··· 2 6 ··· 6 4 ··· 4 4 ··· 4

72 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + + + + + + - + + + + image C1 C2 C2 C2 C2 C2 C2 S3 D5 D6 D6 C4○D4 D10 D10 C4○D12 C4○D20 D4⋊2S3 Q8⋊3S3 S3×D5 C2×S3×D5 D60⋊C2 D6.D10 Dic5.D6 kernel C10.D4⋊S3 D30⋊4C4 C30.Q8 Dic15⋊5C4 C3×C10.D4 Dic3×C20 D30⋊3C4 C10.D4 C4×Dic3 C2×Dic5 C2×C20 C30 C2×Dic3 C2×C12 C10 C6 C10 C10 C2×C4 C22 C2 C2 C2 # reps 1 2 1 1 1 1 1 1 2 2 1 6 4 2 4 24 1 1 2 2 4 4 4

Matrix representation of C10.D4⋊S3 in GL4(𝔽61) generated by

 17 44 0 0 18 0 0 0 0 0 60 0 0 0 0 60
,
 57 54 0 0 46 4 0 0 0 0 52 18 0 0 43 9
,
 2 27 0 0 27 59 0 0 0 0 38 46 0 0 15 23
,
 1 0 0 0 0 1 0 0 0 0 0 60 0 0 1 60
,
 17 45 0 0 18 44 0 0 0 0 0 1 0 0 1 0
`G:=sub<GL(4,GF(61))| [17,18,0,0,44,0,0,0,0,0,60,0,0,0,0,60],[57,46,0,0,54,4,0,0,0,0,52,43,0,0,18,9],[2,27,0,0,27,59,0,0,0,0,38,15,0,0,46,23],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,60,60],[17,18,0,0,45,44,0,0,0,0,0,1,0,0,1,0] >;`

C10.D4⋊S3 in GAP, Magma, Sage, TeX

`C_{10}.D_4\rtimes S_3`
`% in TeX`

`G:=Group("C10.D4:S3");`
`// GroupNames label`

`G:=SmallGroup(480,456);`
`// by ID`

`G=gap.SmallGroup(480,456);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,120,422,142,1356,18822]);`
`// Polycyclic`

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

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