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## G = C3×D10.13D4order 480 = 25·3·5

### Direct product of C3 and D10.13D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C3×D10.13D4
 Chief series C1 — C5 — C10 — C2×C10 — C2×C30 — D5×C2×C6 — D5×C2×C12 — C3×D10.13D4
 Lower central C5 — C2×C10 — C3×D10.13D4
 Upper central C1 — C2×C6 — C3×C4⋊C4

Generators and relations for C3×D10.13D4
G = < a,b,c,d,e | a3=b10=c2=d4=1, e2=b5, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=ece-1=b5c, ede-1=d-1 >

Subgroups: 576 in 156 conjugacy classes, 62 normal (58 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, D4, C23, D5, C10, C12, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, D10, D10, C2×C10, C2×C12, C2×C12, C3×D4, C22×C6, C3×D5, C30, C22.D4, C4×D5, D20, C2×Dic5, C2×C20, C22×D5, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C3×Dic5, C60, C6×D5, C6×D5, C2×C30, C10.D4, D10⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, C3×C22.D4, D5×C12, C3×D20, C6×Dic5, C2×C60, D5×C2×C6, D10.13D4, C3×C10.D4, C3×D10⋊C4, C15×C4⋊C4, D5×C2×C12, C6×D20, C3×D10.13D4
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, C4○D4, D10, C3×D4, C22×C6, C3×D5, C22.D4, C22×D5, C6×D4, C3×C4○D4, C6×D5, C4○D20, D4×D5, Q82D5, C3×C22.D4, D5×C2×C6, D10.13D4, C3×C4○D20, C3×D4×D5, C3×Q82D5, C3×D10.13D4

Smallest permutation representation of C3×D10.13D4
On 240 points
Generators in S240
(1 60 40)(2 51 31)(3 52 32)(4 53 33)(5 54 34)(6 55 35)(7 56 36)(8 57 37)(9 58 38)(10 59 39)(11 225 205)(12 226 206)(13 227 207)(14 228 208)(15 229 209)(16 230 210)(17 221 201)(18 222 202)(19 223 203)(20 224 204)(21 61 41)(22 62 42)(23 63 43)(24 64 44)(25 65 45)(26 66 46)(27 67 47)(28 68 48)(29 69 49)(30 70 50)(71 111 91)(72 112 92)(73 113 93)(74 114 94)(75 115 95)(76 116 96)(77 117 97)(78 118 98)(79 119 99)(80 120 100)(81 121 101)(82 122 102)(83 123 103)(84 124 104)(85 125 105)(86 126 106)(87 127 107)(88 128 108)(89 129 109)(90 130 110)(131 171 151)(132 172 152)(133 173 153)(134 174 154)(135 175 155)(136 176 156)(137 177 157)(138 178 158)(139 179 159)(140 180 160)(141 181 161)(142 182 162)(143 183 163)(144 184 164)(145 185 165)(146 186 166)(147 187 167)(148 188 168)(149 189 169)(150 190 170)(191 231 211)(192 232 212)(193 233 213)(194 234 214)(195 235 215)(196 236 216)(197 237 217)(198 238 218)(199 239 219)(200 240 220)
(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 24)(2 23)(3 22)(4 21)(5 30)(6 29)(7 28)(8 27)(9 26)(10 25)(11 234)(12 233)(13 232)(14 231)(15 240)(16 239)(17 238)(18 237)(19 236)(20 235)(31 43)(32 42)(33 41)(34 50)(35 49)(36 48)(37 47)(38 46)(39 45)(40 44)(51 63)(52 62)(53 61)(54 70)(55 69)(56 68)(57 67)(58 66)(59 65)(60 64)(71 88)(72 87)(73 86)(74 85)(75 84)(76 83)(77 82)(78 81)(79 90)(80 89)(91 108)(92 107)(93 106)(94 105)(95 104)(96 103)(97 102)(98 101)(99 110)(100 109)(111 128)(112 127)(113 126)(114 125)(115 124)(116 123)(117 122)(118 121)(119 130)(120 129)(131 143)(132 142)(133 141)(134 150)(135 149)(136 148)(137 147)(138 146)(139 145)(140 144)(151 163)(152 162)(153 161)(154 170)(155 169)(156 168)(157 167)(158 166)(159 165)(160 164)(171 183)(172 182)(173 181)(174 190)(175 189)(176 188)(177 187)(178 186)(179 185)(180 184)(191 208)(192 207)(193 206)(194 205)(195 204)(196 203)(197 202)(198 201)(199 210)(200 209)(211 228)(212 227)(213 226)(214 225)(215 224)(216 223)(217 222)(218 221)(219 230)(220 229)
(1 85 25 80)(2 86 26 71)(3 87 27 72)(4 88 28 73)(5 89 29 74)(6 90 30 75)(7 81 21 76)(8 82 22 77)(9 83 23 78)(10 84 24 79)(11 180 240 185)(12 171 231 186)(13 172 232 187)(14 173 233 188)(15 174 234 189)(16 175 235 190)(17 176 236 181)(18 177 237 182)(19 178 238 183)(20 179 239 184)(31 106 46 91)(32 107 47 92)(33 108 48 93)(34 109 49 94)(35 110 50 95)(36 101 41 96)(37 102 42 97)(38 103 43 98)(39 104 44 99)(40 105 45 100)(51 126 66 111)(52 127 67 112)(53 128 68 113)(54 129 69 114)(55 130 70 115)(56 121 61 116)(57 122 62 117)(58 123 63 118)(59 124 64 119)(60 125 65 120)(131 191 146 206)(132 192 147 207)(133 193 148 208)(134 194 149 209)(135 195 150 210)(136 196 141 201)(137 197 142 202)(138 198 143 203)(139 199 144 204)(140 200 145 205)(151 211 166 226)(152 212 167 227)(153 213 168 228)(154 214 169 229)(155 215 170 230)(156 216 161 221)(157 217 162 222)(158 218 163 223)(159 219 164 224)(160 220 165 225)
(1 210 6 205)(2 201 7 206)(3 202 8 207)(4 203 9 208)(5 204 10 209)(11 60 16 55)(12 51 17 56)(13 52 18 57)(14 53 19 58)(15 54 20 59)(21 191 26 196)(22 192 27 197)(23 193 28 198)(24 194 29 199)(25 195 30 200)(31 221 36 226)(32 222 37 227)(33 223 38 228)(34 224 39 229)(35 225 40 230)(41 211 46 216)(42 212 47 217)(43 213 48 218)(44 214 49 219)(45 215 50 220)(61 231 66 236)(62 232 67 237)(63 233 68 238)(64 234 69 239)(65 235 70 240)(71 136 76 131)(72 137 77 132)(73 138 78 133)(74 139 79 134)(75 140 80 135)(81 146 86 141)(82 147 87 142)(83 148 88 143)(84 149 89 144)(85 150 90 145)(91 156 96 151)(92 157 97 152)(93 158 98 153)(94 159 99 154)(95 160 100 155)(101 166 106 161)(102 167 107 162)(103 168 108 163)(104 169 109 164)(105 170 110 165)(111 176 116 171)(112 177 117 172)(113 178 118 173)(114 179 119 174)(115 180 120 175)(121 186 126 181)(122 187 127 182)(123 188 128 183)(124 189 129 184)(125 190 130 185)

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

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

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

102 conjugacy classes

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

102 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 D4 D5 C4○D4 D10 C3×D4 C3×D5 C3×C4○D4 C6×D5 C4○D20 C3×C4○D20 D4×D5 Q8⋊2D5 C3×D4×D5 C3×Q8⋊2D5 kernel C3×D10.13D4 C3×C10.D4 C3×D10⋊C4 C15×C4⋊C4 D5×C2×C12 C6×D20 D10.13D4 C10.D4 D10⋊C4 C5×C4⋊C4 C2×C4×D5 C2×D20 C6×D5 C3×C4⋊C4 C30 C2×C12 D10 C4⋊C4 C10 C2×C4 C6 C2 C6 C6 C2 C2 # reps 1 1 3 1 1 1 2 2 6 2 2 2 2 2 4 6 4 4 8 12 8 16 2 2 4 4

Matrix representation of C3×D10.13D4 in GL4(𝔽61) generated by

 1 0 0 0 0 1 0 0 0 0 47 0 0 0 0 47
,
 1 18 0 0 43 43 0 0 0 0 60 0 0 0 0 60
,
 1 18 0 0 0 60 0 0 0 0 1 0 0 0 52 60
,
 31 17 0 0 44 30 0 0 0 0 52 59 0 0 41 9
,
 36 4 0 0 57 25 0 0 0 0 23 39 0 0 13 38
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,47,0,0,0,0,47],[1,43,0,0,18,43,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,18,60,0,0,0,0,1,52,0,0,0,60],[31,44,0,0,17,30,0,0,0,0,52,41,0,0,59,9],[36,57,0,0,4,25,0,0,0,0,23,13,0,0,39,38] >;

C3×D10.13D4 in GAP, Magma, Sage, TeX

C_3\times D_{10}._{13}D_4
% in TeX

G:=Group("C3xD10.13D4");
// GroupNames label

G:=SmallGroup(480,687);
// by ID

G=gap.SmallGroup(480,687);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,176,590,555,268,18822]);
// Polycyclic

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

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