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## G = C3×D4.Dic5order 480 = 25·3·5

### Direct product of C3 and D4.Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×D4.Dic5
 Chief series C1 — C5 — C10 — C20 — C60 — C3×C5⋊2C8 — C6×C5⋊2C8 — C3×D4.Dic5
 Lower central C5 — C10 — C3×D4.Dic5
 Upper central C1 — C12 — C3×C4○D4

Generators and relations for C3×D4.Dic5
G = < a,b,c,d,e | a3=b4=1, c2=d10=b2, e2=d5, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d9 >

Subgroups: 224 in 124 conjugacy classes, 90 normal (24 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, D4, Q8, C10, C10, C12, C12, C2×C6, C15, C2×C8, M4(2), C4○D4, C20, C20, C2×C10, C24, C2×C12, C3×D4, C3×Q8, C30, C30, C8○D4, C52C8, C52C8, C2×C20, C5×D4, C5×Q8, C2×C24, C3×M4(2), C3×C4○D4, C60, C60, C2×C30, C2×C52C8, C4.Dic5, C5×C4○D4, C3×C8○D4, C3×C52C8, C3×C52C8, C2×C60, D4×C15, Q8×C15, D4.Dic5, C6×C52C8, C3×C4.Dic5, C15×C4○D4, C3×D4.Dic5
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, D5, C12, C2×C6, C22×C4, Dic5, D10, C2×C12, C22×C6, C3×D5, C8○D4, C2×Dic5, C22×D5, C22×C12, C3×Dic5, C6×D5, C22×Dic5, C3×C8○D4, C6×Dic5, D5×C2×C6, D4.Dic5, C2×C6×Dic5, C3×D4.Dic5

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

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

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

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

120 conjugacy classes

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

120 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 type + + + + + + - - image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C12 C12 D5 D10 Dic5 Dic5 C3×D5 C8○D4 C6×D5 C3×Dic5 C3×Dic5 C3×C8○D4 D4.Dic5 C3×D4.Dic5 kernel C3×D4.Dic5 C6×C5⋊2C8 C3×C4.Dic5 C15×C4○D4 D4.Dic5 D4×C15 Q8×C15 C2×C5⋊2C8 C4.Dic5 C5×C4○D4 C5×D4 C5×Q8 C3×C4○D4 C2×C12 C3×D4 C3×Q8 C4○D4 C15 C2×C4 D4 Q8 C5 C3 C1 # reps 1 3 3 1 2 6 2 6 6 2 12 4 2 6 6 2 4 4 12 12 4 8 4 8

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

 1 0 0 0 0 1 0 0 0 0 225 0 0 0 0 225
,
 240 0 0 0 0 240 0 0 0 0 64 113 0 0 0 177
,
 240 0 0 0 0 240 0 0 0 0 64 0 0 0 64 177
,
 0 1 0 0 240 190 0 0 0 0 64 0 0 0 0 64
,
 34 240 0 0 193 207 0 0 0 0 8 0 0 0 0 8
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,225,0,0,0,0,225],[240,0,0,0,0,240,0,0,0,0,64,0,0,0,113,177],[240,0,0,0,0,240,0,0,0,0,64,64,0,0,0,177],[0,240,0,0,1,190,0,0,0,0,64,0,0,0,0,64],[34,193,0,0,240,207,0,0,0,0,8,0,0,0,0,8] >;

C3×D4.Dic5 in GAP, Magma, Sage, TeX

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

G:=Group("C3xD4.Dic5");
// GroupNames label

G:=SmallGroup(480,741);
// by ID

G=gap.SmallGroup(480,741);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,168,555,102,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^4=1,c^2=d^10=b^2,e^2=d^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^9>;
// generators/relations

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