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## G = C10×D4⋊S3order 480 = 25·3·5

### Direct product of C10 and D4⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C10×D4⋊S3
 Chief series C1 — C3 — C6 — C12 — C60 — C5×D12 — C10×D12 — C10×D4⋊S3
 Lower central C3 — C6 — C12 — C10×D4⋊S3
 Upper central C1 — C2×C10 — C2×C20 — D4×C10

Generators and relations for C10×D4⋊S3
G = < a,b,c,d,e | a10=b4=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, cd=dc, ece=bc, ede=d-1 >

Subgroups: 420 in 152 conjugacy classes, 66 normal (34 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C22, C22 [×8], C5, S3 [×2], C6, C6 [×2], C6 [×2], C8 [×2], C2×C4, D4 [×2], D4 [×4], C23 [×2], C10, C10 [×2], C10 [×4], C12 [×2], D6 [×4], C2×C6, C2×C6 [×4], C15, C2×C8, D8 [×4], C2×D4, C2×D4, C20 [×2], C2×C10, C2×C10 [×8], C3⋊C8 [×2], D12 [×2], D12, C2×C12, C3×D4 [×2], C3×D4, C22×S3, C22×C6, C5×S3 [×2], C30, C30 [×2], C30 [×2], C2×D8, C40 [×2], C2×C20, C5×D4 [×2], C5×D4 [×4], C22×C10 [×2], C2×C3⋊C8, D4⋊S3 [×4], C2×D12, C6×D4, C60 [×2], S3×C10 [×4], C2×C30, C2×C30 [×4], C2×C40, C5×D8 [×4], D4×C10, D4×C10, C2×D4⋊S3, C5×C3⋊C8 [×2], C5×D12 [×2], C5×D12, C2×C60, D4×C15 [×2], D4×C15, S3×C2×C10, C22×C30, C10×D8, C10×C3⋊C8, C5×D4⋊S3 [×4], C10×D12, D4×C30, C10×D4⋊S3
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×2], C23, C10 [×7], D6 [×3], D8 [×2], C2×D4, C2×C10 [×7], C3⋊D4 [×2], C22×S3, C5×S3, C2×D8, C5×D4 [×2], C22×C10, D4⋊S3 [×2], C2×C3⋊D4, S3×C10 [×3], C5×D8 [×2], D4×C10, C2×D4⋊S3, C5×C3⋊D4 [×2], S3×C2×C10, C10×D8, C5×D4⋊S3 [×2], C10×C3⋊D4, 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)
(1 209 122 61)(2 210 123 62)(3 201 124 63)(4 202 125 64)(5 203 126 65)(6 204 127 66)(7 205 128 67)(8 206 129 68)(9 207 130 69)(10 208 121 70)(11 214 110 43)(12 215 101 44)(13 216 102 45)(14 217 103 46)(15 218 104 47)(16 219 105 48)(17 220 106 49)(18 211 107 50)(19 212 108 41)(20 213 109 42)(21 133 222 200)(22 134 223 191)(23 135 224 192)(24 136 225 193)(25 137 226 194)(26 138 227 195)(27 139 228 196)(28 140 229 197)(29 131 230 198)(30 132 221 199)(31 240 190 100)(32 231 181 91)(33 232 182 92)(34 233 183 93)(35 234 184 94)(36 235 185 95)(37 236 186 96)(38 237 187 97)(39 238 188 98)(40 239 189 99)(51 77 166 146)(52 78 167 147)(53 79 168 148)(54 80 169 149)(55 71 170 150)(56 72 161 141)(57 73 162 142)(58 74 163 143)(59 75 164 144)(60 76 165 145)(81 160 175 111)(82 151 176 112)(83 152 177 113)(84 153 178 114)(85 154 179 115)(86 155 180 116)(87 156 171 117)(88 157 172 118)(89 158 173 119)(90 159 174 120)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 11)(10 12)(21 89)(22 90)(23 81)(24 82)(25 83)(26 84)(27 85)(28 86)(29 87)(30 88)(31 164)(32 165)(33 166)(34 167)(35 168)(36 169)(37 170)(38 161)(39 162)(40 163)(41 205)(42 206)(43 207)(44 208)(45 209)(46 210)(47 201)(48 202)(49 203)(50 204)(51 182)(52 183)(53 184)(54 185)(55 186)(56 187)(57 188)(58 189)(59 190)(60 181)(61 216)(62 217)(63 218)(64 219)(65 220)(66 211)(67 212)(68 213)(69 214)(70 215)(71 236)(72 237)(73 238)(74 239)(75 240)(76 231)(77 232)(78 233)(79 234)(80 235)(91 145)(92 146)(93 147)(94 148)(95 149)(96 150)(97 141)(98 142)(99 143)(100 144)(101 121)(102 122)(103 123)(104 124)(105 125)(106 126)(107 127)(108 128)(109 129)(110 130)(111 135)(112 136)(113 137)(114 138)(115 139)(116 140)(117 131)(118 132)(119 133)(120 134)(151 193)(152 194)(153 195)(154 196)(155 197)(156 198)(157 199)(158 200)(159 191)(160 192)(171 230)(172 221)(173 222)(174 223)(175 224)(176 225)(177 226)(178 227)(179 228)(180 229)
(1 99 160)(2 100 151)(3 91 152)(4 92 153)(5 93 154)(6 94 155)(7 95 156)(8 96 157)(9 97 158)(10 98 159)(11 141 200)(12 142 191)(13 143 192)(14 144 193)(15 145 194)(16 146 195)(17 147 196)(18 148 197)(19 149 198)(20 150 199)(21 214 56)(22 215 57)(23 216 58)(24 217 59)(25 218 60)(26 219 51)(27 220 52)(28 211 53)(29 212 54)(30 213 55)(31 176 210)(32 177 201)(33 178 202)(34 179 203)(35 180 204)(36 171 205)(37 172 206)(38 173 207)(39 174 208)(40 175 209)(41 169 230)(42 170 221)(43 161 222)(44 162 223)(45 163 224)(46 164 225)(47 165 226)(48 166 227)(49 167 228)(50 168 229)(61 189 81)(62 190 82)(63 181 83)(64 182 84)(65 183 85)(66 184 86)(67 185 87)(68 186 88)(69 187 89)(70 188 90)(71 132 109)(72 133 110)(73 134 101)(74 135 102)(75 136 103)(76 137 104)(77 138 105)(78 139 106)(79 140 107)(80 131 108)(111 122 239)(112 123 240)(113 124 231)(114 125 232)(115 126 233)(116 127 234)(117 128 235)(118 129 236)(119 130 237)(120 121 238)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 48)(12 49)(13 50)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 77)(22 78)(23 79)(24 80)(25 71)(26 72)(27 73)(28 74)(29 75)(30 76)(31 87)(32 88)(33 89)(34 90)(35 81)(36 82)(37 83)(38 84)(39 85)(40 86)(51 133)(52 134)(53 135)(54 136)(55 137)(56 138)(57 139)(58 140)(59 131)(60 132)(61 204)(62 205)(63 206)(64 207)(65 208)(66 209)(67 210)(68 201)(69 202)(70 203)(91 157)(92 158)(93 159)(94 160)(95 151)(96 152)(97 153)(98 154)(99 155)(100 156)(101 220)(102 211)(103 212)(104 213)(105 214)(106 215)(107 216)(108 217)(109 218)(110 219)(111 234)(112 235)(113 236)(114 237)(115 238)(116 239)(117 240)(118 231)(119 232)(120 233)(121 126)(122 127)(123 128)(124 129)(125 130)(141 227)(142 228)(143 229)(144 230)(145 221)(146 222)(147 223)(148 224)(149 225)(150 226)(161 195)(162 196)(163 197)(164 198)(165 199)(166 200)(167 191)(168 192)(169 193)(170 194)(171 190)(172 181)(173 182)(174 183)(175 184)(176 185)(177 186)(178 187)(179 188)(180 189)

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

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

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

120 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 5A 5B 5C 5D 6A 6B 6C 6D 6E 6F 6G 8A 8B 8C 8D 10A ··· 10L 10M ··· 10T 10U ··· 10AB 12A 12B 15A 15B 15C 15D 20A ··· 20H 30A ··· 30L 30M ··· 30AB 40A ··· 40P 60A ··· 60H order 1 2 2 2 2 2 2 2 3 4 4 5 5 5 5 6 6 6 6 6 6 6 8 8 8 8 10 ··· 10 10 ··· 10 10 ··· 10 12 12 15 15 15 15 20 ··· 20 30 ··· 30 30 ··· 30 40 ··· 40 60 ··· 60 size 1 1 1 1 4 4 12 12 2 2 2 1 1 1 1 2 2 2 4 4 4 4 6 6 6 6 1 ··· 1 4 ··· 4 12 ··· 12 4 4 2 2 2 2 2 ··· 2 2 ··· 2 4 ··· 4 6 ··· 6 4 ··· 4

120 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C5 C10 C10 C10 C10 S3 D4 D4 D6 D6 D8 C3⋊D4 C3⋊D4 C5×S3 C5×D4 C5×D4 S3×C10 S3×C10 C5×D8 C5×C3⋊D4 C5×C3⋊D4 D4⋊S3 C5×D4⋊S3 kernel C10×D4⋊S3 C10×C3⋊C8 C5×D4⋊S3 C10×D12 D4×C30 C2×D4⋊S3 C2×C3⋊C8 D4⋊S3 C2×D12 C6×D4 D4×C10 C60 C2×C30 C2×C20 C5×D4 C30 C20 C2×C10 C2×D4 C12 C2×C6 C2×C4 D4 C6 C4 C22 C10 C2 # reps 1 1 4 1 1 4 4 16 4 4 1 1 1 1 2 4 2 2 4 4 4 4 8 16 8 8 2 8

Matrix representation of C10×D4⋊S3 in GL4(𝔽241) generated by

 150 0 0 0 0 150 0 0 0 0 143 0 0 0 0 143
,
 1 0 0 0 0 1 0 0 0 0 1 240 0 0 2 240
,
 1 0 0 0 0 1 0 0 0 0 22 230 0 0 22 219
,
 240 1 0 0 240 0 0 0 0 0 1 0 0 0 0 1
,
 240 1 0 0 0 1 0 0 0 0 240 1 0 0 0 1
G:=sub<GL(4,GF(241))| [150,0,0,0,0,150,0,0,0,0,143,0,0,0,0,143],[1,0,0,0,0,1,0,0,0,0,1,2,0,0,240,240],[1,0,0,0,0,1,0,0,0,0,22,22,0,0,230,219],[240,240,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[240,0,0,0,1,1,0,0,0,0,240,0,0,0,1,1] >;

C10×D4⋊S3 in GAP, Magma, Sage, TeX

C_{10}\times D_4\rtimes S_3
% in TeX

G:=Group("C10xD4:S3");
// GroupNames label

G:=SmallGroup(480,810);
// by ID

G=gap.SmallGroup(480,810);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,926,4204,1068,102,15686]);
// Polycyclic

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

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