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## G = C3×D20⋊5C4order 480 = 25·3·5

### Direct product of C3 and D20⋊5C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C3×D20⋊5C4
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×C60 — C6×D20 — C3×D20⋊5C4
 Lower central C5 — C10 — C20 — C3×D20⋊5C4
 Upper central C1 — C2×C6 — C2×C12 — C2×C24

Generators and relations for C3×D205C4
G = < a,b,c,d | a3=b20=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=b3c >

Subgroups: 464 in 100 conjugacy classes, 46 normal (42 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, D4, C23, D5, C10, C12, C12, C2×C6, C2×C6, C15, C4⋊C4, C2×C8, C2×D4, Dic5, C20, D10, C2×C10, C24, C2×C12, C2×C12, C3×D4, C22×C6, C3×D5, C30, D4⋊C4, C40, D20, D20, C2×Dic5, C2×C20, C22×D5, C3×C4⋊C4, C2×C24, C6×D4, C3×Dic5, C60, C6×D5, C2×C30, C4⋊Dic5, C2×C40, C2×D20, C3×D4⋊C4, C120, C3×D20, C3×D20, C6×Dic5, C2×C60, D5×C2×C6, D205C4, C3×C4⋊Dic5, C2×C120, C6×D20, C3×D205C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, D5, C12, C2×C6, C22⋊C4, D8, SD16, D10, C2×C12, C3×D4, C3×D5, D4⋊C4, C4×D5, D20, C5⋊D4, C3×C22⋊C4, C3×D8, C3×SD16, C6×D5, C40⋊C2, D40, D10⋊C4, C3×D4⋊C4, D5×C12, C3×D20, C3×C5⋊D4, D205C4, C3×C40⋊C2, C3×D40, C3×D10⋊C4, C3×D205C4

Smallest permutation representation of C3×D205C4
On 240 points
Generators in S240
(1 173 146)(2 174 147)(3 175 148)(4 176 149)(5 177 150)(6 178 151)(7 179 152)(8 180 153)(9 161 154)(10 162 155)(11 163 156)(12 164 157)(13 165 158)(14 166 159)(15 167 160)(16 168 141)(17 169 142)(18 170 143)(19 171 144)(20 172 145)(21 94 109)(22 95 110)(23 96 111)(24 97 112)(25 98 113)(26 99 114)(27 100 115)(28 81 116)(29 82 117)(30 83 118)(31 84 119)(32 85 120)(33 86 101)(34 87 102)(35 88 103)(36 89 104)(37 90 105)(38 91 106)(39 92 107)(40 93 108)(41 62 232)(42 63 233)(43 64 234)(44 65 235)(45 66 236)(46 67 237)(47 68 238)(48 69 239)(49 70 240)(50 71 221)(51 72 222)(52 73 223)(53 74 224)(54 75 225)(55 76 226)(56 77 227)(57 78 228)(58 79 229)(59 80 230)(60 61 231)(121 213 197)(122 214 198)(123 215 199)(124 216 200)(125 217 181)(126 218 182)(127 219 183)(128 220 184)(129 201 185)(130 202 186)(131 203 187)(132 204 188)(133 205 189)(134 206 190)(135 207 191)(136 208 192)(137 209 193)(138 210 194)(139 211 195)(140 212 196)
(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 48)(2 47)(3 46)(4 45)(5 44)(6 43)(7 42)(8 41)(9 60)(10 59)(11 58)(12 57)(13 56)(14 55)(15 54)(16 53)(17 52)(18 51)(19 50)(20 49)(21 193)(22 192)(23 191)(24 190)(25 189)(26 188)(27 187)(28 186)(29 185)(30 184)(31 183)(32 182)(33 181)(34 200)(35 199)(36 198)(37 197)(38 196)(39 195)(40 194)(61 161)(62 180)(63 179)(64 178)(65 177)(66 176)(67 175)(68 174)(69 173)(70 172)(71 171)(72 170)(73 169)(74 168)(75 167)(76 166)(77 165)(78 164)(79 163)(80 162)(81 130)(82 129)(83 128)(84 127)(85 126)(86 125)(87 124)(88 123)(89 122)(90 121)(91 140)(92 139)(93 138)(94 137)(95 136)(96 135)(97 134)(98 133)(99 132)(100 131)(101 217)(102 216)(103 215)(104 214)(105 213)(106 212)(107 211)(108 210)(109 209)(110 208)(111 207)(112 206)(113 205)(114 204)(115 203)(116 202)(117 201)(118 220)(119 219)(120 218)(141 224)(142 223)(143 222)(144 221)(145 240)(146 239)(147 238)(148 237)(149 236)(150 235)(151 234)(152 233)(153 232)(154 231)(155 230)(156 229)(157 228)(158 227)(159 226)(160 225)
(1 115 59 209)(2 114 60 208)(3 113 41 207)(4 112 42 206)(5 111 43 205)(6 110 44 204)(7 109 45 203)(8 108 46 202)(9 107 47 201)(10 106 48 220)(11 105 49 219)(12 104 50 218)(13 103 51 217)(14 102 52 216)(15 101 53 215)(16 120 54 214)(17 119 55 213)(18 118 56 212)(19 117 57 211)(20 116 58 210)(21 66 187 179)(22 65 188 178)(23 64 189 177)(24 63 190 176)(25 62 191 175)(26 61 192 174)(27 80 193 173)(28 79 194 172)(29 78 195 171)(30 77 196 170)(31 76 197 169)(32 75 198 168)(33 74 199 167)(34 73 200 166)(35 72 181 165)(36 71 182 164)(37 70 183 163)(38 69 184 162)(39 68 185 161)(40 67 186 180)(81 229 138 145)(82 228 139 144)(83 227 140 143)(84 226 121 142)(85 225 122 141)(86 224 123 160)(87 223 124 159)(88 222 125 158)(89 221 126 157)(90 240 127 156)(91 239 128 155)(92 238 129 154)(93 237 130 153)(94 236 131 152)(95 235 132 151)(96 234 133 150)(97 233 134 149)(98 232 135 148)(99 231 136 147)(100 230 137 146)

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

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

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

138 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 5A 5B 6A ··· 6F 6G 6H 6I 6J 8A 8B 8C 8D 10A ··· 10F 12A 12B 12C 12D 12E 12F 12G 12H 15A 15B 15C 15D 20A ··· 20H 24A ··· 24H 30A ··· 30L 40A ··· 40P 60A ··· 60P 120A ··· 120AF order 1 2 2 2 2 2 3 3 4 4 4 4 5 5 6 ··· 6 6 6 6 6 8 8 8 8 10 ··· 10 12 12 12 12 12 12 12 12 15 15 15 15 20 ··· 20 24 ··· 24 30 ··· 30 40 ··· 40 60 ··· 60 120 ··· 120 size 1 1 1 1 20 20 1 1 2 2 20 20 2 2 1 ··· 1 20 20 20 20 2 2 2 2 2 ··· 2 2 2 2 2 20 20 20 20 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

138 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 D4 D4 D5 D8 SD16 D10 C3×D4 C3×D4 C3×D5 C4×D5 C5⋊D4 D20 C3×D8 C3×SD16 C6×D5 C40⋊C2 D40 D5×C12 C3×C5⋊D4 C3×D20 C3×C40⋊C2 C3×D40 kernel C3×D20⋊5C4 C3×C4⋊Dic5 C2×C120 C6×D20 D20⋊5C4 C3×D20 C4⋊Dic5 C2×C40 C2×D20 D20 C60 C2×C30 C2×C24 C30 C30 C2×C12 C20 C2×C10 C2×C8 C12 C12 C2×C6 C10 C10 C2×C4 C6 C6 C4 C4 C22 C2 C2 # reps 1 1 1 1 2 4 2 2 2 8 1 1 2 2 2 2 2 2 4 4 4 4 4 4 4 8 8 8 8 8 16 16

Matrix representation of C3×D205C4 in GL4(𝔽241) generated by

 225 0 0 0 0 225 0 0 0 0 15 0 0 0 0 15
,
 240 0 0 0 0 240 0 0 0 0 41 44 0 0 78 119
,
 1 0 0 0 40 240 0 0 0 0 156 78 0 0 204 85
,
 232 133 0 0 32 9 0 0 0 0 116 210 0 0 162 125
G:=sub<GL(4,GF(241))| [225,0,0,0,0,225,0,0,0,0,15,0,0,0,0,15],[240,0,0,0,0,240,0,0,0,0,41,78,0,0,44,119],[1,40,0,0,0,240,0,0,0,0,156,204,0,0,78,85],[232,32,0,0,133,9,0,0,0,0,116,162,0,0,210,125] >;

C3×D205C4 in GAP, Magma, Sage, TeX

C_3\times D_{20}\rtimes_5C_4
% in TeX

G:=Group("C3xD20:5C4");
// GroupNames label

G:=SmallGroup(480,99);
// by ID

G=gap.SmallGroup(480,99);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,197,260,1683,136,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^20=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=d*b*d^-1=b^-1,d*c*d^-1=b^3*c>;
// generators/relations

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