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## G = C5×C4⋊Dic3order 240 = 24·3·5

### Direct product of C5 and C4⋊Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C5×C4⋊Dic3
 Chief series C1 — C3 — C6 — C2×C6 — C2×C30 — C10×Dic3 — C5×C4⋊Dic3
 Lower central C3 — C6 — C5×C4⋊Dic3
 Upper central C1 — C2×C10 — C2×C20

Generators and relations for C5×C4⋊Dic3
G = < a,b,c,d | a5=b4=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Smallest permutation representation of C5×C4⋊Dic3
Regular action on 240 points
Generators in S240
(1 58 46 34 22)(2 59 47 35 23)(3 60 48 36 24)(4 55 43 31 19)(5 56 44 32 20)(6 57 45 33 21)(7 232 220 208 196)(8 233 221 209 197)(9 234 222 210 198)(10 229 217 205 193)(11 230 218 206 194)(12 231 219 207 195)(13 61 49 37 25)(14 62 50 38 26)(15 63 51 39 27)(16 64 52 40 28)(17 65 53 41 29)(18 66 54 42 30)(67 115 103 91 79)(68 116 104 92 80)(69 117 105 93 81)(70 118 106 94 82)(71 119 107 95 83)(72 120 108 96 84)(73 122 109 97 85)(74 123 110 98 86)(75 124 111 99 87)(76 125 112 100 88)(77 126 113 101 89)(78 121 114 102 90)(127 175 163 151 139)(128 176 164 152 140)(129 177 165 153 141)(130 178 166 154 142)(131 179 167 155 143)(132 180 168 156 144)(133 181 169 157 145)(134 182 170 158 146)(135 183 171 159 147)(136 184 172 160 148)(137 185 173 161 149)(138 186 174 162 150)(187 235 223 211 199)(188 236 224 212 200)(189 237 225 213 201)(190 238 226 214 202)(191 239 227 215 203)(192 240 228 216 204)
(1 77 17 70)(2 78 18 71)(3 73 13 72)(4 74 14 67)(5 75 15 68)(6 76 16 69)(7 176 236 184)(8 177 237 185)(9 178 238 186)(10 179 239 181)(11 180 240 182)(12 175 235 183)(19 86 26 79)(20 87 27 80)(21 88 28 81)(22 89 29 82)(23 90 30 83)(24 85 25 84)(31 98 38 91)(32 99 39 92)(33 100 40 93)(34 101 41 94)(35 102 42 95)(36 97 37 96)(43 110 50 103)(44 111 51 104)(45 112 52 105)(46 113 53 106)(47 114 54 107)(48 109 49 108)(55 123 62 115)(56 124 63 116)(57 125 64 117)(58 126 65 118)(59 121 66 119)(60 122 61 120)(127 187 135 195)(128 188 136 196)(129 189 137 197)(130 190 138 198)(131 191 133 193)(132 192 134 194)(139 199 147 207)(140 200 148 208)(141 201 149 209)(142 202 150 210)(143 203 145 205)(144 204 146 206)(151 211 159 219)(152 212 160 220)(153 213 161 221)(154 214 162 222)(155 215 157 217)(156 216 158 218)(163 223 171 231)(164 224 172 232)(165 225 173 233)(166 226 174 234)(167 227 169 229)(168 228 170 230)
(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)
(1 131 4 128)(2 130 5 127)(3 129 6 132)(7 126 10 123)(8 125 11 122)(9 124 12 121)(13 137 16 134)(14 136 17 133)(15 135 18 138)(19 140 22 143)(20 139 23 142)(21 144 24 141)(25 149 28 146)(26 148 29 145)(27 147 30 150)(31 152 34 155)(32 151 35 154)(33 156 36 153)(37 161 40 158)(38 160 41 157)(39 159 42 162)(43 164 46 167)(44 163 47 166)(45 168 48 165)(49 173 52 170)(50 172 53 169)(51 171 54 174)(55 176 58 179)(56 175 59 178)(57 180 60 177)(61 185 64 182)(62 184 65 181)(63 183 66 186)(67 188 70 191)(68 187 71 190)(69 192 72 189)(73 197 76 194)(74 196 77 193)(75 195 78 198)(79 200 82 203)(80 199 83 202)(81 204 84 201)(85 209 88 206)(86 208 89 205)(87 207 90 210)(91 212 94 215)(92 211 95 214)(93 216 96 213)(97 221 100 218)(98 220 101 217)(99 219 102 222)(103 224 106 227)(104 223 107 226)(105 228 108 225)(109 233 112 230)(110 232 113 229)(111 231 114 234)(115 236 118 239)(116 235 119 238)(117 240 120 237)

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

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

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

90 conjugacy classes

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

90 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 C4 C5 C10 C10 C20 S3 D4 Q8 Dic3 D6 Dic6 D12 C5×S3 C5×D4 C5×Q8 C5×Dic3 S3×C10 C5×Dic6 C5×D12 kernel C5×C4⋊Dic3 C10×Dic3 C2×C60 C60 C4⋊Dic3 C2×Dic3 C2×C12 C12 C2×C20 C30 C30 C20 C2×C10 C10 C10 C2×C4 C6 C6 C4 C22 C2 C2 # reps 1 2 1 4 4 8 4 16 1 1 1 2 1 2 2 4 4 4 8 4 8 8

Matrix representation of C5×C4⋊Dic3 in GL3(𝔽61) generated by

 1 0 0 0 34 0 0 0 34
,
 60 0 0 0 38 15 0 46 23
,
 60 0 0 0 1 1 0 60 0
,
 11 0 0 0 40 38 0 59 21
G:=sub<GL(3,GF(61))| [1,0,0,0,34,0,0,0,34],[60,0,0,0,38,46,0,15,23],[60,0,0,0,1,60,0,1,0],[11,0,0,0,40,59,0,38,21] >;

C5×C4⋊Dic3 in GAP, Magma, Sage, TeX

C_5\times C_4\rtimes {\rm Dic}_3
% in TeX

G:=Group("C5xC4:Dic3");
// GroupNames label

G:=SmallGroup(240,58);
// by ID

G=gap.SmallGroup(240,58);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-3,120,505,247,5765]);
// Polycyclic

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

Export

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