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## G = C2×S3×Dic10order 480 = 25·3·5

### Direct product of C2, S3 and Dic10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C2×S3×Dic10
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — S3×Dic5 — C2×S3×Dic5 — C2×S3×Dic10
 Lower central C15 — C30 — C2×S3×Dic10
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×S3×Dic10
G = < a,b,c,d,e | a2=b3=c2=d20=1, e2=d10, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 1276 in 312 conjugacy classes, 132 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, Q8, C23, C10, C10, C10, Dic3, Dic3, C12, C12, D6, C2×C6, C15, C22×C4, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, C2×C10, Dic6, C4×S3, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C5×S3, C30, C30, C22×Q8, Dic10, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C2×Dic6, S3×C2×C4, S3×C2×C4, S3×Q8, C6×Q8, C5×Dic3, C3×Dic5, Dic15, C60, S3×C10, C2×C30, C2×Dic10, C2×Dic10, C22×Dic5, C22×C20, C2×S3×Q8, S3×Dic5, C15⋊Q8, C3×Dic10, C6×Dic5, S3×C20, C10×Dic3, Dic30, C2×Dic15, C2×C60, S3×C2×C10, C22×Dic10, S3×Dic10, C2×S3×Dic5, C2×C15⋊Q8, C6×Dic10, S3×C2×C20, C2×Dic30, C2×S3×Dic10
Quotients: C1, C2, C22, S3, Q8, C23, D5, D6, C2×Q8, C24, D10, C22×S3, C22×Q8, Dic10, C22×D5, S3×Q8, S3×C23, S3×D5, C2×Dic10, C23×D5, C2×S3×Q8, C2×S3×D5, C22×Dic10, S3×Dic10, C22×S3×D5, C2×S3×Dic10

Smallest permutation representation of C2×S3×Dic10
On 240 points
Generators in S240
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(11 40)(12 21)(13 22)(14 23)(15 24)(16 25)(17 26)(18 27)(19 28)(20 29)(41 112)(42 113)(43 114)(44 115)(45 116)(46 117)(47 118)(48 119)(49 120)(50 101)(51 102)(52 103)(53 104)(54 105)(55 106)(56 107)(57 108)(58 109)(59 110)(60 111)(61 203)(62 204)(63 205)(64 206)(65 207)(66 208)(67 209)(68 210)(69 211)(70 212)(71 213)(72 214)(73 215)(74 216)(75 217)(76 218)(77 219)(78 220)(79 201)(80 202)(81 127)(82 128)(83 129)(84 130)(85 131)(86 132)(87 133)(88 134)(89 135)(90 136)(91 137)(92 138)(93 139)(94 140)(95 121)(96 122)(97 123)(98 124)(99 125)(100 126)(141 228)(142 229)(143 230)(144 231)(145 232)(146 233)(147 234)(148 235)(149 236)(150 237)(151 238)(152 239)(153 240)(154 221)(155 222)(156 223)(157 224)(158 225)(159 226)(160 227)(161 191)(162 192)(163 193)(164 194)(165 195)(166 196)(167 197)(168 198)(169 199)(170 200)(171 181)(172 182)(173 183)(174 184)(175 185)(176 186)(177 187)(178 188)(179 189)(180 190)
(1 199 220)(2 200 201)(3 181 202)(4 182 203)(5 183 204)(6 184 205)(7 185 206)(8 186 207)(9 187 208)(10 188 209)(11 189 210)(12 190 211)(13 191 212)(14 192 213)(15 193 214)(16 194 215)(17 195 216)(18 196 217)(19 197 218)(20 198 219)(21 180 69)(22 161 70)(23 162 71)(24 163 72)(25 164 73)(26 165 74)(27 166 75)(28 167 76)(29 168 77)(30 169 78)(31 170 79)(32 171 80)(33 172 61)(34 173 62)(35 174 63)(36 175 64)(37 176 65)(38 177 66)(39 178 67)(40 179 68)(41 121 154)(42 122 155)(43 123 156)(44 124 157)(45 125 158)(46 126 159)(47 127 160)(48 128 141)(49 129 142)(50 130 143)(51 131 144)(52 132 145)(53 133 146)(54 134 147)(55 135 148)(56 136 149)(57 137 150)(58 138 151)(59 139 152)(60 140 153)(81 227 118)(82 228 119)(83 229 120)(84 230 101)(85 231 102)(86 232 103)(87 233 104)(88 234 105)(89 235 106)(90 236 107)(91 237 108)(92 238 109)(93 239 110)(94 240 111)(95 221 112)(96 222 113)(97 223 114)(98 224 115)(99 225 116)(100 226 117)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 131)(42 132)(43 133)(44 134)(45 135)(46 136)(47 137)(48 138)(49 139)(50 140)(51 121)(52 122)(53 123)(54 124)(55 125)(56 126)(57 127)(58 128)(59 129)(60 130)(61 162)(62 163)(63 164)(64 165)(65 166)(66 167)(67 168)(68 169)(69 170)(70 171)(71 172)(72 173)(73 174)(74 175)(75 176)(76 177)(77 178)(78 179)(79 180)(80 161)(81 108)(82 109)(83 110)(84 111)(85 112)(86 113)(87 114)(88 115)(89 116)(90 117)(91 118)(92 119)(93 120)(94 101)(95 102)(96 103)(97 104)(98 105)(99 106)(100 107)(141 151)(142 152)(143 153)(144 154)(145 155)(146 156)(147 157)(148 158)(149 159)(150 160)(181 212)(182 213)(183 214)(184 215)(185 216)(186 217)(187 218)(188 219)(189 220)(190 201)(191 202)(192 203)(193 204)(194 205)(195 206)(196 207)(197 208)(198 209)(199 210)(200 211)(221 231)(222 232)(223 233)(224 234)(225 235)(226 236)(227 237)(228 238)(229 239)(230 240)
(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)
(1 151 11 141)(2 150 12 160)(3 149 13 159)(4 148 14 158)(5 147 15 157)(6 146 16 156)(7 145 17 155)(8 144 18 154)(9 143 19 153)(10 142 20 152)(21 227 31 237)(22 226 32 236)(23 225 33 235)(24 224 34 234)(25 223 35 233)(26 222 36 232)(27 221 37 231)(28 240 38 230)(29 239 39 229)(30 238 40 228)(41 186 51 196)(42 185 52 195)(43 184 53 194)(44 183 54 193)(45 182 55 192)(46 181 56 191)(47 200 57 190)(48 199 58 189)(49 198 59 188)(50 197 60 187)(61 89 71 99)(62 88 72 98)(63 87 73 97)(64 86 74 96)(65 85 75 95)(66 84 76 94)(67 83 77 93)(68 82 78 92)(69 81 79 91)(70 100 80 90)(101 167 111 177)(102 166 112 176)(103 165 113 175)(104 164 114 174)(105 163 115 173)(106 162 116 172)(107 161 117 171)(108 180 118 170)(109 179 119 169)(110 178 120 168)(121 207 131 217)(122 206 132 216)(123 205 133 215)(124 204 134 214)(125 203 135 213)(126 202 136 212)(127 201 137 211)(128 220 138 210)(129 219 139 209)(130 218 140 208)

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

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

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

78 conjugacy classes

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

78 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + - + + + + + + + + - - + + + - image C1 C2 C2 C2 C2 C2 C2 S3 Q8 D5 D6 D6 D6 D10 D10 D10 D10 Dic10 S3×Q8 S3×D5 C2×S3×D5 C2×S3×D5 S3×Dic10 kernel C2×S3×Dic10 S3×Dic10 C2×S3×Dic5 C2×C15⋊Q8 C6×Dic10 S3×C2×C20 C2×Dic30 C2×Dic10 S3×C10 S3×C2×C4 Dic10 C2×Dic5 C2×C20 C4×S3 C2×Dic3 C2×C12 C22×S3 D6 C10 C2×C4 C4 C22 C2 # reps 1 8 2 2 1 1 1 1 4 2 4 2 1 8 2 2 2 16 2 2 4 2 8

Matrix representation of C2×S3×Dic10 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 1 0 0 0 0 0 0 1
,
 1 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 0 60 0 0 0 0 1 60
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 50 0 0 0 0 0 11 11 0 0 0 0 0 0 17 60 0 0 0 0 45 1 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 60 59 0 0 0 0 1 1 0 0 0 0 0 0 33 38 0 0 0 0 42 28 0 0 0 0 0 0 60 0 0 0 0 0 0 60

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,60,60],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[50,11,0,0,0,0,0,11,0,0,0,0,0,0,17,45,0,0,0,0,60,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[60,1,0,0,0,0,59,1,0,0,0,0,0,0,33,42,0,0,0,0,38,28,0,0,0,0,0,0,60,0,0,0,0,0,0,60] >;

C2×S3×Dic10 in GAP, Magma, Sage, TeX

C_2\times S_3\times {\rm Dic}_{10}
% in TeX

G:=Group("C2xS3xDic10");
// GroupNames label

G:=SmallGroup(480,1078);
// by ID

G=gap.SmallGroup(480,1078);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,675,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^2=d^20=1,e^2=d^10,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,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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