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## G = C3×C4⋊Dic5order 240 = 24·3·5

### Direct product of C3 and C4⋊Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×C4⋊Dic5
 Chief series C1 — C5 — C10 — C2×C10 — C2×C30 — C6×Dic5 — C3×C4⋊Dic5
 Lower central C5 — C10 — C3×C4⋊Dic5
 Upper central C1 — C2×C6 — C2×C12

Generators and relations for C3×C4⋊Dic5
G = < a,b,c,d | a3=b4=c10=1, d2=c5, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Smallest permutation representation of C3×C4⋊Dic5
Regular action on 240 points
Generators in S240
(1 52 32)(2 53 33)(3 54 34)(4 55 35)(5 56 36)(6 57 37)(7 58 38)(8 59 39)(9 60 40)(10 51 31)(11 228 208)(12 229 209)(13 230 210)(14 221 201)(15 222 202)(16 223 203)(17 224 204)(18 225 205)(19 226 206)(20 227 207)(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)(81 124 101)(82 125 102)(83 126 103)(84 127 104)(85 128 105)(86 129 106)(87 130 107)(88 121 108)(89 122 109)(90 123 110)(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 87 27 72)(2 88 28 73)(3 89 29 74)(4 90 30 75)(5 81 21 76)(6 82 22 77)(7 83 23 78)(8 84 24 79)(9 85 25 80)(10 86 26 71)(11 178 238 188)(12 179 239 189)(13 180 240 190)(14 171 231 181)(15 172 232 182)(16 173 233 183)(17 174 234 184)(18 175 235 185)(19 176 236 186)(20 177 237 187)(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)(51 129 66 111)(52 130 67 112)(53 121 68 113)(54 122 69 114)(55 123 70 115)(56 124 61 116)(57 125 62 117)(58 126 63 118)(59 127 64 119)(60 128 65 120)(131 191 141 201)(132 192 142 202)(133 193 143 203)(134 194 144 204)(135 195 145 205)(136 196 146 206)(137 197 147 207)(138 198 148 208)(139 199 149 209)(140 200 150 210)(151 211 161 221)(152 212 162 222)(153 213 163 223)(154 214 164 224)(155 215 165 225)(156 216 166 226)(157 217 167 227)(158 218 168 228)(159 219 169 229)(160 220 170 230)
(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)
(1 133 6 138)(2 132 7 137)(3 131 8 136)(4 140 9 135)(5 139 10 134)(11 130 16 125)(12 129 17 124)(13 128 18 123)(14 127 19 122)(15 126 20 121)(21 149 26 144)(22 148 27 143)(23 147 28 142)(24 146 29 141)(25 145 30 150)(31 154 36 159)(32 153 37 158)(33 152 38 157)(34 151 39 156)(35 160 40 155)(41 169 46 164)(42 168 47 163)(43 167 48 162)(44 166 49 161)(45 165 50 170)(51 174 56 179)(52 173 57 178)(53 172 58 177)(54 171 59 176)(55 180 60 175)(61 189 66 184)(62 188 67 183)(63 187 68 182)(64 186 69 181)(65 185 70 190)(71 194 76 199)(72 193 77 198)(73 192 78 197)(74 191 79 196)(75 200 80 195)(81 209 86 204)(82 208 87 203)(83 207 88 202)(84 206 89 201)(85 205 90 210)(91 214 96 219)(92 213 97 218)(93 212 98 217)(94 211 99 216)(95 220 100 215)(101 229 106 224)(102 228 107 223)(103 227 108 222)(104 226 109 221)(105 225 110 230)(111 234 116 239)(112 233 117 238)(113 232 118 237)(114 231 119 236)(115 240 120 235)

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

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

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

78 conjugacy classes

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

78 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + - + - + - + image C1 C2 C2 C3 C4 C6 C6 C12 D4 Q8 D5 Dic5 D10 C3×D4 C3×Q8 C3×D5 Dic10 D20 C3×Dic5 C6×D5 C3×Dic10 C3×D20 kernel C3×C4⋊Dic5 C6×Dic5 C2×C60 C4⋊Dic5 C60 C2×Dic5 C2×C20 C20 C30 C30 C2×C12 C12 C2×C6 C10 C10 C2×C4 C6 C6 C4 C22 C2 C2 # reps 1 2 1 2 4 4 2 8 1 1 2 4 2 2 2 4 4 4 8 4 8 8

Matrix representation of C3×C4⋊Dic5 in GL3(𝔽61) generated by

 47 0 0 0 13 0 0 0 13
,
 60 0 0 0 29 54 0 7 32
,
 60 0 0 0 17 60 0 1 0
,
 50 0 0 0 37 28 0 47 24
G:=sub<GL(3,GF(61))| [47,0,0,0,13,0,0,0,13],[60,0,0,0,29,7,0,54,32],[60,0,0,0,17,1,0,60,0],[50,0,0,0,37,47,0,28,24] >;

C3×C4⋊Dic5 in GAP, Magma, Sage, TeX

C_3\times C_4\rtimes {\rm Dic}_5
% in TeX

G:=Group("C3xC4:Dic5");
// GroupNames label

G:=SmallGroup(240,42);
// by ID

G=gap.SmallGroup(240,42);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-5,72,313,151,6917]);
// Polycyclic

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

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