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## G = C4×D5×Dic3order 480 = 25·3·5

### Direct product of C4, D5 and Dic3

Series: Derived Chief Lower central Upper central

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

Generators and relations for C4×D5×Dic3
G = < a,b,c,d,e | a4=b5=c2=d6=1, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 796 in 216 conjugacy classes, 104 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, C23, D5, C10, C10, Dic3, Dic3, C12, C12, C2×C6, C2×C6, C15, C42, C22×C4, Dic5, Dic5, C20, C20, D10, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C3×D5, C30, C30, C2×C42, C4×D5, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C4×Dic3, C4×Dic3, C22×Dic3, C22×C12, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, C2×C30, C4×Dic5, C4×C20, C2×C4×D5, C2×C4×D5, C2×C4×Dic3, D5×Dic3, D5×C12, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, D5×C2×C6, D5×C42, Dic3×Dic5, Dic3×C20, C4×Dic15, C2×D5×Dic3, D5×C2×C12, C4×D5×Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, Dic3, D6, C42, C22×C4, D10, C4×S3, C2×Dic3, C22×S3, C2×C42, C4×D5, C22×D5, C4×Dic3, S3×C2×C4, C22×Dic3, S3×D5, C2×C4×D5, C2×C4×Dic3, D5×Dic3, C2×S3×D5, D5×C42, C4×S3×D5, C2×D5×Dic3, C4×D5×Dic3

Smallest permutation representation of C4×D5×Dic3
On 240 points
Generators in S240
(1 115 16 106)(2 116 17 107)(3 117 18 108)(4 118 13 103)(5 119 14 104)(6 120 15 105)(7 214 229 169)(8 215 230 170)(9 216 231 171)(10 211 232 172)(11 212 233 173)(12 213 234 174)(19 152 82 113)(20 153 83 114)(21 154 84 109)(22 155 79 110)(23 156 80 111)(24 151 81 112)(25 148 88 73)(26 149 89 74)(27 150 90 75)(28 145 85 76)(29 146 86 77)(30 147 87 78)(31 144 94 64)(32 139 95 65)(33 140 96 66)(34 141 91 61)(35 142 92 62)(36 143 93 63)(37 160 100 71)(38 161 101 72)(39 162 102 67)(40 157 97 68)(41 158 98 69)(42 159 99 70)(43 210 195 135)(44 205 196 136)(45 206 197 137)(46 207 198 138)(47 208 193 133)(48 209 194 134)(49 200 191 131)(50 201 192 132)(51 202 187 127)(52 203 188 128)(53 204 189 129)(54 199 190 130)(55 221 181 121)(56 222 182 122)(57 217 183 123)(58 218 184 124)(59 219 185 125)(60 220 186 126)(163 238 178 223)(164 239 179 224)(165 240 180 225)(166 235 175 226)(167 236 176 227)(168 237 177 228)
(1 101 83 95 29)(2 102 84 96 30)(3 97 79 91 25)(4 98 80 92 26)(5 99 81 93 27)(6 100 82 94 28)(7 51 196 227 60)(8 52 197 228 55)(9 53 198 223 56)(10 54 193 224 57)(11 49 194 225 58)(12 50 195 226 59)(13 41 23 35 89)(14 42 24 36 90)(15 37 19 31 85)(16 38 20 32 86)(17 39 21 33 87)(18 40 22 34 88)(43 235 185 234 192)(44 236 186 229 187)(45 237 181 230 188)(46 238 182 231 189)(47 239 183 232 190)(48 240 184 233 191)(61 148 117 68 110)(62 149 118 69 111)(63 150 119 70 112)(64 145 120 71 113)(65 146 115 72 114)(66 147 116 67 109)(73 108 157 155 141)(74 103 158 156 142)(75 104 159 151 143)(76 105 160 152 144)(77 106 161 153 139)(78 107 162 154 140)(121 170 128 206 177)(122 171 129 207 178)(123 172 130 208 179)(124 173 131 209 180)(125 174 132 210 175)(126 169 127 205 176)(133 164 217 211 199)(134 165 218 212 200)(135 166 219 213 201)(136 167 220 214 202)(137 168 221 215 203)(138 163 222 216 204)
(1 29)(2 30)(3 25)(4 26)(5 27)(6 28)(13 89)(14 90)(15 85)(16 86)(17 87)(18 88)(31 37)(32 38)(33 39)(34 40)(35 41)(36 42)(43 235)(44 236)(45 237)(46 238)(47 239)(48 240)(49 58)(50 59)(51 60)(52 55)(53 56)(54 57)(61 68)(62 69)(63 70)(64 71)(65 72)(66 67)(73 108)(74 103)(75 104)(76 105)(77 106)(78 107)(91 97)(92 98)(93 99)(94 100)(95 101)(96 102)(115 146)(116 147)(117 148)(118 149)(119 150)(120 145)(121 128)(122 129)(123 130)(124 131)(125 132)(126 127)(133 164)(134 165)(135 166)(136 167)(137 168)(138 163)(139 161)(140 162)(141 157)(142 158)(143 159)(144 160)(175 210)(176 205)(177 206)(178 207)(179 208)(180 209)(181 188)(182 189)(183 190)(184 191)(185 192)(186 187)(193 224)(194 225)(195 226)(196 227)(197 228)(198 223)(199 217)(200 218)(201 219)(202 220)(203 221)(204 222)
(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)
(1 227 4 224)(2 226 5 223)(3 225 6 228)(7 80 10 83)(8 79 11 82)(9 84 12 81)(13 239 16 236)(14 238 17 235)(15 237 18 240)(19 230 22 233)(20 229 23 232)(21 234 24 231)(25 194 28 197)(26 193 29 196)(27 198 30 195)(31 188 34 191)(32 187 35 190)(33 192 36 189)(37 181 40 184)(38 186 41 183)(39 185 42 182)(43 90 46 87)(44 89 47 86)(45 88 48 85)(49 94 52 91)(50 93 53 96)(51 92 54 95)(55 97 58 100)(56 102 59 99)(57 101 60 98)(61 200 64 203)(62 199 65 202)(63 204 66 201)(67 219 70 222)(68 218 71 221)(69 217 72 220)(73 209 76 206)(74 208 77 205)(75 207 78 210)(103 179 106 176)(104 178 107 175)(105 177 108 180)(109 213 112 216)(110 212 113 215)(111 211 114 214)(115 167 118 164)(116 166 119 163)(117 165 120 168)(121 157 124 160)(122 162 125 159)(123 161 126 158)(127 142 130 139)(128 141 131 144)(129 140 132 143)(133 146 136 149)(134 145 137 148)(135 150 138 147)(151 171 154 174)(152 170 155 173)(153 169 156 172)

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

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

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

96 conjugacy classes

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

96 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + - + + + + + + - + image C1 C2 C2 C2 C2 C2 C4 C4 S3 D5 Dic3 D6 D6 D6 D10 D10 C4×S3 C4×D5 C4×D5 S3×D5 D5×Dic3 C2×S3×D5 C4×S3×D5 kernel C4×D5×Dic3 Dic3×Dic5 Dic3×C20 C4×Dic15 C2×D5×Dic3 D5×C2×C12 D5×Dic3 D5×C12 C2×C4×D5 C4×Dic3 C4×D5 C2×Dic5 C2×C20 C22×D5 C2×Dic3 C2×C12 D10 Dic3 C12 C2×C4 C4 C22 C2 # reps 1 2 1 1 2 1 16 8 1 2 4 1 1 1 4 2 8 16 8 2 4 2 8

Matrix representation of C4×D5×Dic3 in GL4(𝔽61) generated by

 50 0 0 0 0 50 0 0 0 0 1 0 0 0 0 1
,
 44 1 0 0 16 60 0 0 0 0 1 0 0 0 0 1
,
 60 60 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 60 0 0 0 0 60 0 0 0 0 60 15 0 0 12 2
,
 11 0 0 0 0 11 0 0 0 0 11 18 0 0 0 50
G:=sub<GL(4,GF(61))| [50,0,0,0,0,50,0,0,0,0,1,0,0,0,0,1],[44,16,0,0,1,60,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,60,1,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,60,12,0,0,15,2],[11,0,0,0,0,11,0,0,0,0,11,0,0,0,18,50] >;

C4×D5×Dic3 in GAP, Magma, Sage, TeX

C_4\times D_5\times {\rm Dic}_3
% in TeX

G:=Group("C4xD5xDic3");
// GroupNames label

G:=SmallGroup(480,467);
// by ID

G=gap.SmallGroup(480,467);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,64,100,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^5=c^2=d^6=1,e^2=d^3,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

׿
×
𝔽