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## G = C3×Dic5.D4order 480 = 25·3·5

### Direct product of C3 and Dic5.D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C3×Dic5.D4
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C6×Dic5 — C3×C22.F5 — C3×Dic5.D4
 Lower central C5 — C10 — C2×C10 — C3×Dic5.D4
 Upper central C1 — C6 — C2×C6 — C2×C12

Generators and relations for C3×Dic5.D4
G = < a,b,c,d,e | a3=b10=1, c2=d4=b5, e2=c, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, dbd-1=ebe-1=b3, dcd-1=b5c, ce=ec, ede-1=cd3 >

Subgroups: 248 in 76 conjugacy classes, 32 normal (24 characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C6, C8, C2×C4, C2×C4, Q8, C10, C10, C12, C2×C6, C15, M4(2), C2×Q8, Dic5, Dic5, C20, C2×C10, C24, C2×C12, C2×C12, C3×Q8, C30, C30, C4.10D4, C5⋊C8, Dic10, C2×Dic5, C2×C20, C3×M4(2), C6×Q8, C3×Dic5, C3×Dic5, C60, C2×C30, C22.F5, C2×Dic10, C3×C4.10D4, C3×C5⋊C8, C3×Dic10, C6×Dic5, C2×C60, Dic5.D4, C3×C22.F5, C6×Dic10, C3×Dic5.D4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C12, C2×C6, C22⋊C4, F5, C2×C12, C3×D4, C4.10D4, C2×F5, C3×C22⋊C4, C3×F5, C22⋊F5, C3×C4.10D4, C6×F5, Dic5.D4, C3×C22⋊F5, C3×Dic5.D4

Smallest permutation representation of C3×Dic5.D4
On 240 points
Generators in S240
(1 41 21)(2 42 22)(3 43 23)(4 44 24)(5 45 25)(6 46 26)(7 47 27)(8 48 28)(9 49 29)(10 50 30)(11 51 31)(12 52 32)(13 53 33)(14 54 34)(15 55 35)(16 56 36)(17 57 37)(18 58 38)(19 59 39)(20 60 40)(61 101 81)(62 102 82)(63 103 83)(64 104 84)(65 105 85)(66 106 86)(67 107 87)(68 108 88)(69 109 89)(70 110 90)(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)(121 161 141)(122 162 142)(123 163 143)(124 164 144)(125 165 145)(126 166 146)(127 167 147)(128 168 148)(129 169 149)(130 170 150)(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)(181 221 201)(182 222 202)(183 223 203)(184 224 204)(185 225 205)(186 226 206)(187 227 207)(188 228 208)(189 229 209)(190 230 210)(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 14 6 19)(2 13 7 18)(3 12 8 17)(4 11 9 16)(5 20 10 15)(21 34 26 39)(22 33 27 38)(23 32 28 37)(24 31 29 36)(25 40 30 35)(41 54 46 59)(42 53 47 58)(43 52 48 57)(44 51 49 56)(45 60 50 55)(61 79 66 74)(62 78 67 73)(63 77 68 72)(64 76 69 71)(65 75 70 80)(81 99 86 94)(82 98 87 93)(83 97 88 92)(84 96 89 91)(85 95 90 100)(101 119 106 114)(102 118 107 113)(103 117 108 112)(104 116 109 111)(105 115 110 120)(121 134 126 139)(122 133 127 138)(123 132 128 137)(124 131 129 136)(125 140 130 135)(141 154 146 159)(142 153 147 158)(143 152 148 157)(144 151 149 156)(145 160 150 155)(161 174 166 179)(162 173 167 178)(163 172 168 177)(164 171 169 176)(165 180 170 175)(181 199 186 194)(182 198 187 193)(183 197 188 192)(184 196 189 191)(185 195 190 200)(201 219 206 214)(202 218 207 213)(203 217 208 212)(204 216 209 211)(205 215 210 220)(221 239 226 234)(222 238 227 233)(223 237 228 232)(224 236 229 231)(225 235 230 240)
(1 197 19 188 6 192 14 183)(2 194 18 181 7 199 13 186)(3 191 17 184 8 196 12 189)(4 198 16 187 9 193 11 182)(5 195 15 190 10 200 20 185)(21 217 39 208 26 212 34 203)(22 214 38 201 27 219 33 206)(23 211 37 204 28 216 32 209)(24 218 36 207 29 213 31 202)(25 215 35 210 30 220 40 205)(41 237 59 228 46 232 54 223)(42 234 58 221 47 239 53 226)(43 231 57 224 48 236 52 229)(44 238 56 227 49 233 51 222)(45 235 55 230 50 240 60 225)(61 137 74 123 66 132 79 128)(62 134 73 126 67 139 78 121)(63 131 72 129 68 136 77 124)(64 138 71 122 69 133 76 127)(65 135 80 125 70 140 75 130)(81 157 94 143 86 152 99 148)(82 154 93 146 87 159 98 141)(83 151 92 149 88 156 97 144)(84 158 91 142 89 153 96 147)(85 155 100 145 90 160 95 150)(101 177 114 163 106 172 119 168)(102 174 113 166 107 179 118 161)(103 171 112 169 108 176 117 164)(104 178 111 162 109 173 116 167)(105 175 120 165 110 180 115 170)
(1 132 14 128 6 137 19 123)(2 139 13 121 7 134 18 126)(3 136 12 124 8 131 17 129)(4 133 11 127 9 138 16 122)(5 140 20 130 10 135 15 125)(21 152 34 148 26 157 39 143)(22 159 33 141 27 154 38 146)(23 156 32 144 28 151 37 149)(24 153 31 147 29 158 36 142)(25 160 40 150 30 155 35 145)(41 172 54 168 46 177 59 163)(42 179 53 161 47 174 58 166)(43 176 52 164 48 171 57 169)(44 173 51 167 49 178 56 162)(45 180 60 170 50 175 55 165)(61 197 79 188 66 192 74 183)(62 194 78 181 67 199 73 186)(63 191 77 184 68 196 72 189)(64 198 76 187 69 193 71 182)(65 195 75 190 70 200 80 185)(81 217 99 208 86 212 94 203)(82 214 98 201 87 219 93 206)(83 211 97 204 88 216 92 209)(84 218 96 207 89 213 91 202)(85 215 95 210 90 220 100 205)(101 237 119 228 106 232 114 223)(102 234 118 221 107 239 113 226)(103 231 117 224 108 236 112 229)(104 238 116 227 109 233 111 222)(105 235 115 230 110 240 120 225)

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

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

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

57 conjugacy classes

 class 1 2A 2B 3A 3B 4A 4B 4C 4D 5 6A 6B 6C 6D 8A 8B 8C 8D 10A 10B 10C 12A 12B 12C 12D 12E 12F 12G 12H 15A 15B 20A 20B 20C 20D 24A ··· 24H 30A ··· 30F 60A ··· 60H order 1 2 2 3 3 4 4 4 4 5 6 6 6 6 8 8 8 8 10 10 10 12 12 12 12 12 12 12 12 15 15 20 20 20 20 24 ··· 24 30 ··· 30 60 ··· 60 size 1 1 2 1 1 4 10 10 20 4 1 1 2 2 20 20 20 20 4 4 4 4 4 10 10 10 10 20 20 4 4 4 4 4 4 20 ··· 20 4 ··· 4 4 ··· 4

57 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 4 4 4 4 4 4 4 4 4 4 type + + + + + - + + - image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 D4 C3×D4 F5 C4.10D4 C2×F5 C3×F5 C22⋊F5 C3×C4.10D4 C6×F5 Dic5.D4 C3×C22⋊F5 C3×Dic5.D4 kernel C3×Dic5.D4 C3×C22.F5 C6×Dic10 Dic5.D4 C6×Dic5 C2×C60 C22.F5 C2×Dic10 C2×Dic5 C2×C20 C3×Dic5 Dic5 C2×C12 C15 C2×C6 C2×C4 C6 C5 C22 C3 C2 C1 # reps 1 2 1 2 2 2 4 2 4 4 2 4 1 1 1 2 2 2 2 4 4 8

Matrix representation of C3×Dic5.D4 in GL4(𝔽241) generated by

 15 0 0 0 0 15 0 0 0 0 15 0 0 0 0 15
,
 0 240 0 0 1 190 0 0 0 0 1 190 0 0 51 51
,
 232 167 0 0 190 9 0 0 0 0 232 167 0 0 190 9
,
 0 0 197 238 0 0 3 44 67 96 0 0 139 174 0 0
,
 0 0 1 0 0 0 0 1 232 167 0 0 190 9 0 0
G:=sub<GL(4,GF(241))| [15,0,0,0,0,15,0,0,0,0,15,0,0,0,0,15],[0,1,0,0,240,190,0,0,0,0,1,51,0,0,190,51],[232,190,0,0,167,9,0,0,0,0,232,190,0,0,167,9],[0,0,67,139,0,0,96,174,197,3,0,0,238,44,0,0],[0,0,232,190,0,0,167,9,1,0,0,0,0,1,0,0] >;

C3×Dic5.D4 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_5.D_4
% in TeX

G:=Group("C3xDic5.D4");
// GroupNames label

G:=SmallGroup(480,285);
// by ID

G=gap.SmallGroup(480,285);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,365,344,850,136,2524,9414,1595]);
// Polycyclic

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

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