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## G = C10×D4⋊2S3order 480 = 25·3·5

### Direct product of C10 and D4⋊2S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C10×D4⋊2S3
 Chief series C1 — C3 — C6 — C30 — S3×C10 — S3×C2×C10 — S3×C2×C20 — C10×D4⋊2S3
 Lower central C3 — C6 — C10×D4⋊2S3
 Upper central C1 — C2×C10 — D4×C10

Generators and relations for C10×D42S3
G = < a,b,c,d,e | a10=b4=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >

Subgroups: 612 in 328 conjugacy classes, 178 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×2], C4 [×6], C22, C22 [×4], C22 [×8], C5, S3 [×2], C6, C6 [×2], C6 [×4], C2×C4, C2×C4 [×15], D4 [×4], D4 [×8], Q8 [×4], C23 [×2], C23, C10, C10 [×2], C10 [×6], Dic3 [×6], C12 [×2], D6 [×2], D6 [×2], C2×C6, C2×C6 [×4], C2×C6 [×4], C15, C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×8], C20 [×2], C20 [×6], C2×C10, C2×C10 [×4], C2×C10 [×8], Dic6 [×4], C4×S3 [×4], C2×Dic3, C2×Dic3 [×10], C3⋊D4 [×8], C2×C12, C3×D4 [×4], C22×S3, C22×C6 [×2], C5×S3 [×2], C30, C30 [×2], C30 [×4], C2×C4○D4, C2×C20, C2×C20 [×15], C5×D4 [×4], C5×D4 [×8], C5×Q8 [×4], C22×C10 [×2], C22×C10, C2×Dic6, S3×C2×C4, D42S3 [×8], C22×Dic3 [×2], C2×C3⋊D4 [×2], C6×D4, C5×Dic3 [×6], C60 [×2], S3×C10 [×2], S3×C10 [×2], C2×C30, C2×C30 [×4], C2×C30 [×4], C22×C20 [×3], D4×C10, D4×C10 [×2], Q8×C10, C5×C4○D4 [×8], C2×D42S3, C5×Dic6 [×4], S3×C20 [×4], C10×Dic3, C10×Dic3 [×10], C5×C3⋊D4 [×8], C2×C60, D4×C15 [×4], S3×C2×C10, C22×C30 [×2], C10×C4○D4, C10×Dic6, S3×C2×C20, C5×D42S3 [×8], Dic3×C2×C10 [×2], C10×C3⋊D4 [×2], D4×C30, C10×D42S3
Quotients: C1, C2 [×15], C22 [×35], C5, S3, C23 [×15], C10 [×15], D6 [×7], C4○D4 [×2], C24, C2×C10 [×35], C22×S3 [×7], C5×S3, C2×C4○D4, C22×C10 [×15], D42S3 [×2], S3×C23, S3×C10 [×7], C5×C4○D4 [×2], C23×C10, C2×D42S3, S3×C2×C10 [×7], C10×C4○D4, C5×D42S3 [×2], S3×C22×C10, C10×D42S3

Smallest permutation representation of C10×D42S3
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)
(1 141 81 155)(2 142 82 156)(3 143 83 157)(4 144 84 158)(5 145 85 159)(6 146 86 160)(7 147 87 151)(8 148 88 152)(9 149 89 153)(10 150 90 154)(11 230 36 181)(12 221 37 182)(13 222 38 183)(14 223 39 184)(15 224 40 185)(16 225 31 186)(17 226 32 187)(18 227 33 188)(19 228 34 189)(20 229 35 190)(21 210 53 214)(22 201 54 215)(23 202 55 216)(24 203 56 217)(25 204 57 218)(26 205 58 219)(27 206 59 220)(28 207 60 211)(29 208 51 212)(30 209 52 213)(41 200 234 174)(42 191 235 175)(43 192 236 176)(44 193 237 177)(45 194 238 178)(46 195 239 179)(47 196 240 180)(48 197 231 171)(49 198 232 172)(50 199 233 173)(61 170 110 121)(62 161 101 122)(63 162 102 123)(64 163 103 124)(65 164 104 125)(66 165 105 126)(67 166 106 127)(68 167 107 128)(69 168 108 129)(70 169 109 130)(71 115 100 131)(72 116 91 132)(73 117 92 133)(74 118 93 134)(75 119 94 135)(76 120 95 136)(77 111 96 137)(78 112 97 138)(79 113 98 139)(80 114 99 140)
(1 160)(2 151)(3 152)(4 153)(5 154)(6 155)(7 156)(8 157)(9 158)(10 159)(11 186)(12 187)(13 188)(14 189)(15 190)(16 181)(17 182)(18 183)(19 184)(20 185)(21 219)(22 220)(23 211)(24 212)(25 213)(26 214)(27 215)(28 216)(29 217)(30 218)(31 230)(32 221)(33 222)(34 223)(35 224)(36 225)(37 226)(38 227)(39 228)(40 229)(41 179)(42 180)(43 171)(44 172)(45 173)(46 174)(47 175)(48 176)(49 177)(50 178)(51 203)(52 204)(53 205)(54 206)(55 207)(56 208)(57 209)(58 210)(59 201)(60 202)(61 126)(62 127)(63 128)(64 129)(65 130)(66 121)(67 122)(68 123)(69 124)(70 125)(71 136)(72 137)(73 138)(74 139)(75 140)(76 131)(77 132)(78 133)(79 134)(80 135)(81 146)(82 147)(83 148)(84 149)(85 150)(86 141)(87 142)(88 143)(89 144)(90 145)(91 111)(92 112)(93 113)(94 114)(95 115)(96 116)(97 117)(98 118)(99 119)(100 120)(101 166)(102 167)(103 168)(104 169)(105 170)(106 161)(107 162)(108 163)(109 164)(110 165)(191 240)(192 231)(193 232)(194 233)(195 234)(196 235)(197 236)(198 237)(199 238)(200 239)
(1 94 103)(2 95 104)(3 96 105)(4 97 106)(5 98 107)(6 99 108)(7 100 109)(8 91 110)(9 92 101)(10 93 102)(11 236 51)(12 237 52)(13 238 53)(14 239 54)(15 240 55)(16 231 56)(17 232 57)(18 233 58)(19 234 59)(20 235 60)(21 38 45)(22 39 46)(23 40 47)(24 31 48)(25 32 49)(26 33 50)(27 34 41)(28 35 42)(29 36 43)(30 37 44)(61 88 72)(62 89 73)(63 90 74)(64 81 75)(65 82 76)(66 83 77)(67 84 78)(68 85 79)(69 86 80)(70 87 71)(111 165 157)(112 166 158)(113 167 159)(114 168 160)(115 169 151)(116 170 152)(117 161 153)(118 162 154)(119 163 155)(120 164 156)(121 148 132)(122 149 133)(123 150 134)(124 141 135)(125 142 136)(126 143 137)(127 144 138)(128 145 139)(129 146 140)(130 147 131)(171 217 225)(172 218 226)(173 219 227)(174 220 228)(175 211 229)(176 212 230)(177 213 221)(178 214 222)(179 215 223)(180 216 224)(181 192 208)(182 193 209)(183 194 210)(184 195 201)(185 196 202)(186 197 203)(187 198 204)(188 199 205)(189 200 206)(190 191 207)
(1 215)(2 216)(3 217)(4 218)(5 219)(6 220)(7 211)(8 212)(9 213)(10 214)(11 170)(12 161)(13 162)(14 163)(15 164)(16 165)(17 166)(18 167)(19 168)(20 169)(21 150)(22 141)(23 142)(24 143)(25 144)(26 145)(27 146)(28 147)(29 148)(30 149)(31 126)(32 127)(33 128)(34 129)(35 130)(36 121)(37 122)(38 123)(39 124)(40 125)(41 140)(42 131)(43 132)(44 133)(45 134)(46 135)(47 136)(48 137)(49 138)(50 139)(51 152)(52 153)(53 154)(54 155)(55 156)(56 157)(57 158)(58 159)(59 160)(60 151)(61 181)(62 182)(63 183)(64 184)(65 185)(66 186)(67 187)(68 188)(69 189)(70 190)(71 191)(72 192)(73 193)(74 194)(75 195)(76 196)(77 197)(78 198)(79 199)(80 200)(81 201)(82 202)(83 203)(84 204)(85 205)(86 206)(87 207)(88 208)(89 209)(90 210)(91 176)(92 177)(93 178)(94 179)(95 180)(96 171)(97 172)(98 173)(99 174)(100 175)(101 221)(102 222)(103 223)(104 224)(105 225)(106 226)(107 227)(108 228)(109 229)(110 230)(111 231)(112 232)(113 233)(114 234)(115 235)(116 236)(117 237)(118 238)(119 239)(120 240)

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

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

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

150 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 5A 5B 5C 5D 6A 6B 6C 6D 6E 6F 6G 10A ··· 10L 10M ··· 10AB 10AC ··· 10AJ 12A 12B 15A 15B 15C 15D 20A ··· 20H 20I ··· 20X 20Y ··· 20AN 30A ··· 30L 30M ··· 30AB 60A ··· 60H order 1 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 5 5 5 5 6 6 6 6 6 6 6 10 ··· 10 10 ··· 10 10 ··· 10 12 12 15 15 15 15 20 ··· 20 20 ··· 20 20 ··· 20 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 2 2 6 6 2 2 2 3 3 3 3 6 6 6 6 1 1 1 1 2 2 2 4 4 4 4 1 ··· 1 2 ··· 2 6 ··· 6 4 4 2 2 2 2 2 ··· 2 3 ··· 3 6 ··· 6 2 ··· 2 4 ··· 4 4 ··· 4

150 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 C10 S3 D6 D6 D6 C4○D4 C5×S3 S3×C10 S3×C10 S3×C10 C5×C4○D4 D4⋊2S3 C5×D4⋊2S3 kernel C10×D4⋊2S3 C10×Dic6 S3×C2×C20 C5×D4⋊2S3 Dic3×C2×C10 C10×C3⋊D4 D4×C30 C2×D4⋊2S3 C2×Dic6 S3×C2×C4 D4⋊2S3 C22×Dic3 C2×C3⋊D4 C6×D4 D4×C10 C2×C20 C5×D4 C22×C10 C30 C2×D4 C2×C4 D4 C23 C6 C10 C2 # reps 1 1 1 8 2 2 1 4 4 4 32 8 8 4 1 1 4 2 4 4 4 16 8 16 2 8

Matrix representation of C10×D42S3 in GL5(𝔽61)

 41 0 0 0 0 0 20 0 0 0 0 0 20 0 0 0 0 0 9 0 0 0 0 0 9
,
 60 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 3 0 0 0 40 60
,
 60 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 3 0 0 0 0 60
,
 1 0 0 0 0 0 0 1 0 0 0 60 60 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 19 31 0 0 0 12 42 0 0 0 0 0 50 28 0 0 0 48 11

G:=sub<GL(5,GF(61))| [41,0,0,0,0,0,20,0,0,0,0,0,20,0,0,0,0,0,9,0,0,0,0,0,9],[60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,40,0,0,0,3,60],[60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,3,60],[1,0,0,0,0,0,0,60,0,0,0,1,60,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,19,12,0,0,0,31,42,0,0,0,0,0,50,48,0,0,0,28,11] >;

C10×D42S3 in GAP, Magma, Sage, TeX

C_{10}\times D_4\rtimes_2S_3
% in TeX

G:=Group("C10xD4:2S3");
// GroupNames label

G:=SmallGroup(480,1155);
// by ID

G=gap.SmallGroup(480,1155);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-3,436,2467,633,15686]);
// 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=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations

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