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## G = C2×D20⋊S3order 480 = 25·3·5

### Direct product of C2 and D20⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C2×D20⋊S3
 Chief series C1 — C5 — C15 — C30 — C6×D5 — D5×Dic3 — C2×D5×Dic3 — C2×D20⋊S3
 Lower central C15 — C30 — C2×D20⋊S3
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×D20⋊S3
G = < a,b,c,d,e | a2=b20=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe=b9, cd=dc, ece=b18c, ede=d-1 >

Subgroups: 1596 in 328 conjugacy classes, 116 normal (24 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C6, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C15, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, D10, C2×C10, Dic6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C3×D5, D15, C30, C30, C2×C4○D4, C4×D5, D20, D20, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C22×D5, C2×Dic6, S3×C2×C4, D42S3, C22×Dic3, C2×C3⋊D4, C6×D4, C5×Dic3, Dic15, C60, C6×D5, C6×D5, D30, D30, C2×C30, C2×C4×D5, C2×D20, C2×D20, Q82D5, Q8×C10, C2×D42S3, D5×Dic3, C3⋊D20, C3×D20, C5×Dic6, C10×Dic3, C4×D15, C2×Dic15, C2×C60, D5×C2×C6, C22×D15, C2×Q82D5, D20⋊S3, C2×D5×Dic3, C2×C3⋊D20, C6×D20, C10×Dic6, C2×C4×D15, C2×D20⋊S3
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, C24, D10, C22×S3, C2×C4○D4, C22×D5, D42S3, S3×C23, S3×D5, Q82D5, C23×D5, C2×D42S3, C2×S3×D5, C2×Q82D5, D20⋊S3, C22×S3×D5, C2×D20⋊S3

Smallest permutation representation of C2×D20⋊S3
On 240 points
Generators in S240
(1 133)(2 134)(3 135)(4 136)(5 137)(6 138)(7 139)(8 140)(9 121)(10 122)(11 123)(12 124)(13 125)(14 126)(15 127)(16 128)(17 129)(18 130)(19 131)(20 132)(21 161)(22 162)(23 163)(24 164)(25 165)(26 166)(27 167)(28 168)(29 169)(30 170)(31 171)(32 172)(33 173)(34 174)(35 175)(36 176)(37 177)(38 178)(39 179)(40 180)(41 206)(42 207)(43 208)(44 209)(45 210)(46 211)(47 212)(48 213)(49 214)(50 215)(51 216)(52 217)(53 218)(54 219)(55 220)(56 201)(57 202)(58 203)(59 204)(60 205)(61 87)(62 88)(63 89)(64 90)(65 91)(66 92)(67 93)(68 94)(69 95)(70 96)(71 97)(72 98)(73 99)(74 100)(75 81)(76 82)(77 83)(78 84)(79 85)(80 86)(101 142)(102 143)(103 144)(104 145)(105 146)(106 147)(107 148)(108 149)(109 150)(110 151)(111 152)(112 153)(113 154)(114 155)(115 156)(116 157)(117 158)(118 159)(119 160)(120 141)(181 231)(182 232)(183 233)(184 234)(185 235)(186 236)(187 237)(188 238)(189 239)(190 240)(191 221)(192 222)(193 223)(194 224)(195 225)(196 226)(197 227)(198 228)(199 229)(200 230)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 28)(35 40)(36 39)(37 38)(41 50)(42 49)(43 48)(44 47)(45 46)(51 60)(52 59)(53 58)(54 57)(55 56)(61 70)(62 69)(63 68)(64 67)(65 66)(71 80)(72 79)(73 78)(74 77)(75 76)(81 82)(83 100)(84 99)(85 98)(86 97)(87 96)(88 95)(89 94)(90 93)(91 92)(101 114)(102 113)(103 112)(104 111)(105 110)(106 109)(107 108)(115 120)(116 119)(117 118)(121 124)(122 123)(125 140)(126 139)(127 138)(128 137)(129 136)(130 135)(131 134)(132 133)(141 156)(142 155)(143 154)(144 153)(145 152)(146 151)(147 150)(148 149)(157 160)(158 159)(161 174)(162 173)(163 172)(164 171)(165 170)(166 169)(167 168)(175 180)(176 179)(177 178)(181 192)(182 191)(183 190)(184 189)(185 188)(186 187)(193 200)(194 199)(195 198)(196 197)(201 220)(202 219)(203 218)(204 217)(205 216)(206 215)(207 214)(208 213)(209 212)(210 211)(221 232)(222 231)(223 230)(224 229)(225 228)(226 227)(233 240)(234 239)(235 238)(236 237)
(1 66 197)(2 67 198)(3 68 199)(4 69 200)(5 70 181)(6 71 182)(7 72 183)(8 73 184)(9 74 185)(10 75 186)(11 76 187)(12 77 188)(13 78 189)(14 79 190)(15 80 191)(16 61 192)(17 62 193)(18 63 194)(19 64 195)(20 65 196)(21 142 59)(22 143 60)(23 144 41)(24 145 42)(25 146 43)(26 147 44)(27 148 45)(28 149 46)(29 150 47)(30 151 48)(31 152 49)(32 153 50)(33 154 51)(34 155 52)(35 156 53)(36 157 54)(37 158 55)(38 159 56)(39 160 57)(40 141 58)(81 236 122)(82 237 123)(83 238 124)(84 239 125)(85 240 126)(86 221 127)(87 222 128)(88 223 129)(89 224 130)(90 225 131)(91 226 132)(92 227 133)(93 228 134)(94 229 135)(95 230 136)(96 231 137)(97 232 138)(98 233 139)(99 234 140)(100 235 121)(101 204 161)(102 205 162)(103 206 163)(104 207 164)(105 208 165)(106 209 166)(107 210 167)(108 211 168)(109 212 169)(110 213 170)(111 214 171)(112 215 172)(113 216 173)(114 217 174)(115 218 175)(116 219 176)(117 220 177)(118 201 178)(119 202 179)(120 203 180)
(1 33)(2 22)(3 31)(4 40)(5 29)(6 38)(7 27)(8 36)(9 25)(10 34)(11 23)(12 32)(13 21)(14 30)(15 39)(16 28)(17 37)(18 26)(19 35)(20 24)(41 76)(42 65)(43 74)(44 63)(45 72)(46 61)(47 70)(48 79)(49 68)(50 77)(51 66)(52 75)(53 64)(54 73)(55 62)(56 71)(57 80)(58 69)(59 78)(60 67)(81 217)(82 206)(83 215)(84 204)(85 213)(86 202)(87 211)(88 220)(89 209)(90 218)(91 207)(92 216)(93 205)(94 214)(95 203)(96 212)(97 201)(98 210)(99 219)(100 208)(101 239)(102 228)(103 237)(104 226)(105 235)(106 224)(107 233)(108 222)(109 231)(110 240)(111 229)(112 238)(113 227)(114 236)(115 225)(116 234)(117 223)(118 232)(119 221)(120 230)(121 165)(122 174)(123 163)(124 172)(125 161)(126 170)(127 179)(128 168)(129 177)(130 166)(131 175)(132 164)(133 173)(134 162)(135 171)(136 180)(137 169)(138 178)(139 167)(140 176)(141 200)(142 189)(143 198)(144 187)(145 196)(146 185)(147 194)(148 183)(149 192)(150 181)(151 190)(152 199)(153 188)(154 197)(155 186)(156 195)(157 184)(158 193)(159 182)(160 191)

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

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

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

66 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 6A 6B 6C 6D 6E 6F 6G 10A ··· 10F 12A 12B 15A 15B 20A 20B 20C 20D 20E ··· 20L 30A ··· 30F 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 6 6 6 6 6 6 6 10 ··· 10 12 12 15 15 20 20 20 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 10 10 10 10 30 30 2 2 2 6 6 6 6 15 15 15 15 2 2 2 2 2 20 20 20 20 2 ··· 2 4 4 4 4 4 4 4 4 12 ··· 12 4 ··· 4 4 ··· 4

66 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 type + + + + + + + + + + + + + + + - + + + + image C1 C2 C2 C2 C2 C2 C2 S3 D5 D6 D6 D6 C4○D4 D10 D10 D10 D4⋊2S3 S3×D5 Q8⋊2D5 C2×S3×D5 C2×S3×D5 D20⋊S3 kernel C2×D20⋊S3 D20⋊S3 C2×D5×Dic3 C2×C3⋊D20 C6×D20 C10×Dic6 C2×C4×D15 C2×D20 C2×Dic6 D20 C2×C20 C22×D5 C30 Dic6 C2×Dic3 C2×C12 C10 C2×C4 C6 C4 C22 C2 # reps 1 8 2 2 1 1 1 1 2 4 1 2 4 8 4 2 2 2 4 4 2 8

Matrix representation of C2×D20⋊S3 in GL6(𝔽61)

 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 0 1 0 0 0 0 60 18 0 0 0 0 0 0 45 46 0 0 0 0 9 16
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 43 1 0 0 0 0 43 18 0 0 0 0 0 0 45 46 0 0 0 0 17 16
,
 0 1 0 0 0 0 60 60 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 1
,
 39 48 0 0 0 0 9 22 0 0 0 0 0 0 43 18 0 0 0 0 60 18 0 0 0 0 0 0 54 43 0 0 0 0 23 7

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,1,18,0,0,0,0,0,0,45,9,0,0,0,0,46,16],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,43,43,0,0,0,0,1,18,0,0,0,0,0,0,45,17,0,0,0,0,46,16],[0,60,0,0,0,0,1,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[39,9,0,0,0,0,48,22,0,0,0,0,0,0,43,60,0,0,0,0,18,18,0,0,0,0,0,0,54,23,0,0,0,0,43,7] >;

C2×D20⋊S3 in GAP, Magma, Sage, TeX

C_2\times D_{20}\rtimes S_3
% in TeX

G:=Group("C2xD20:S3");
// GroupNames label

G:=SmallGroup(480,1075);
// by ID

G=gap.SmallGroup(480,1075);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,675,346,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^20=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,e*b*e=b^9,c*d=d*c,e*c*e=b^18*c,e*d*e=d^-1>;
// generators/relations

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