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## G = C3×D20⋊6C4order 480 = 25·3·5

### Direct product of C3 and D20⋊6C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C3×D20⋊6C4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C60 — C6×D20 — C3×D20⋊6C4
 Lower central C5 — C10 — C20 — C3×D20⋊6C4
 Upper central C1 — C2×C6 — C2×C12 — C3×C4⋊C4

Generators and relations for C3×D206C4
G = < a,b,c,d | a3=b20=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b11, dcd-1=b15c >

Subgroups: 432 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, C20, C20, D10, C2×C10, C24, C2×C12, C2×C12, C3×D4, C22×C6, C3×D5, C30, D4⋊C4, C52C8, D20, D20, C2×C20, C2×C20, C22×D5, C3×C4⋊C4, C2×C24, C6×D4, C60, C60, C6×D5, C2×C30, C2×C52C8, C5×C4⋊C4, C2×D20, C3×D4⋊C4, C3×C52C8, C3×D20, C3×D20, C2×C60, C2×C60, D5×C2×C6, D206C4, C6×C52C8, C15×C4⋊C4, C6×D20, C3×D206C4
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, D10⋊C4, D4⋊D5, Q8⋊D5, C3×D4⋊C4, D5×C12, C3×D20, C3×C5⋊D4, D206C4, C3×D10⋊C4, C3×D4⋊D5, C3×Q8⋊D5, C3×D206C4

Smallest permutation representation of C3×D206C4
On 240 points
Generators in S240
(1 91 46)(2 92 47)(3 93 48)(4 94 49)(5 95 50)(6 96 51)(7 97 52)(8 98 53)(9 99 54)(10 100 55)(11 81 56)(12 82 57)(13 83 58)(14 84 59)(15 85 60)(16 86 41)(17 87 42)(18 88 43)(19 89 44)(20 90 45)(21 106 61)(22 107 62)(23 108 63)(24 109 64)(25 110 65)(26 111 66)(27 112 67)(28 113 68)(29 114 69)(30 115 70)(31 116 71)(32 117 72)(33 118 73)(34 119 74)(35 120 75)(36 101 76)(37 102 77)(38 103 78)(39 104 79)(40 105 80)(121 206 176)(122 207 177)(123 208 178)(124 209 179)(125 210 180)(126 211 161)(127 212 162)(128 213 163)(129 214 164)(130 215 165)(131 216 166)(132 217 167)(133 218 168)(134 219 169)(135 220 170)(136 201 171)(137 202 172)(138 203 173)(139 204 174)(140 205 175)(141 231 196)(142 232 197)(143 233 198)(144 234 199)(145 235 200)(146 236 181)(147 237 182)(148 238 183)(149 239 184)(150 240 185)(151 221 186)(152 222 187)(153 223 188)(154 224 189)(155 225 190)(156 226 191)(157 227 192)(158 228 193)(159 229 194)(160 230 195)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 40)(22 39)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(41 50)(42 49)(43 48)(44 47)(45 46)(51 60)(52 59)(53 58)(54 57)(55 56)(61 80)(62 79)(63 78)(64 77)(65 76)(66 75)(67 74)(68 73)(69 72)(70 71)(81 100)(82 99)(83 98)(84 97)(85 96)(86 95)(87 94)(88 93)(89 92)(90 91)(101 110)(102 109)(103 108)(104 107)(105 106)(111 120)(112 119)(113 118)(114 117)(115 116)(121 135)(122 134)(123 133)(124 132)(125 131)(126 130)(127 129)(136 140)(137 139)(141 145)(142 144)(146 160)(147 159)(148 158)(149 157)(150 156)(151 155)(152 154)(161 165)(162 164)(166 180)(167 179)(168 178)(169 177)(170 176)(171 175)(172 174)(181 195)(182 194)(183 193)(184 192)(185 191)(186 190)(187 189)(196 200)(197 199)(201 205)(202 204)(206 220)(207 219)(208 218)(209 217)(210 216)(211 215)(212 214)(221 225)(222 224)(226 240)(227 239)(228 238)(229 237)(230 236)(231 235)(232 234)
(1 151 21 136)(2 142 22 127)(3 153 23 138)(4 144 24 129)(5 155 25 140)(6 146 26 131)(7 157 27 122)(8 148 28 133)(9 159 29 124)(10 150 30 135)(11 141 31 126)(12 152 32 137)(13 143 33 128)(14 154 34 139)(15 145 35 130)(16 156 36 121)(17 147 37 132)(18 158 38 123)(19 149 39 134)(20 160 40 125)(41 191 76 176)(42 182 77 167)(43 193 78 178)(44 184 79 169)(45 195 80 180)(46 186 61 171)(47 197 62 162)(48 188 63 173)(49 199 64 164)(50 190 65 175)(51 181 66 166)(52 192 67 177)(53 183 68 168)(54 194 69 179)(55 185 70 170)(56 196 71 161)(57 187 72 172)(58 198 73 163)(59 189 74 174)(60 200 75 165)(81 231 116 211)(82 222 117 202)(83 233 118 213)(84 224 119 204)(85 235 120 215)(86 226 101 206)(87 237 102 217)(88 228 103 208)(89 239 104 219)(90 230 105 210)(91 221 106 201)(92 232 107 212)(93 223 108 203)(94 234 109 214)(95 225 110 205)(96 236 111 216)(97 227 112 207)(98 238 113 218)(99 229 114 209)(100 240 115 220)

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

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

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

102 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 ··· 20L 24A ··· 24H 30A ··· 30L 60A ··· 60X 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 60 ··· 60 size 1 1 1 1 20 20 1 1 2 2 4 4 2 2 1 ··· 1 20 20 20 20 10 10 10 10 2 ··· 2 2 2 2 2 4 4 4 4 2 2 2 2 4 ··· 4 10 ··· 10 2 ··· 2 4 ··· 4

102 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 4 4 4 4 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 D20 C5⋊D4 C3×D8 C3×SD16 C6×D5 D5×C12 C3×D20 C3×C5⋊D4 D4⋊D5 Q8⋊D5 C3×D4⋊D5 C3×Q8⋊D5 kernel C3×D20⋊6C4 C6×C5⋊2C8 C15×C4⋊C4 C6×D20 D20⋊6C4 C3×D20 C2×C5⋊2C8 C5×C4⋊C4 C2×D20 D20 C60 C2×C30 C3×C4⋊C4 C30 C30 C2×C12 C20 C2×C10 C4⋊C4 C12 C12 C2×C6 C10 C10 C2×C4 C4 C4 C22 C6 C6 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 2 2 4 4

Matrix representation of C3×D206C4 in GL4(𝔽241) generated by

 225 0 0 0 0 225 0 0 0 0 1 0 0 0 0 1
,
 1 2 0 0 240 240 0 0 0 0 52 1 0 0 240 0
,
 1 0 0 0 240 240 0 0 0 0 52 1 0 0 189 189
,
 0 22 0 0 11 0 0 0 0 0 200 156 0 0 85 41
G:=sub<GL(4,GF(241))| [225,0,0,0,0,225,0,0,0,0,1,0,0,0,0,1],[1,240,0,0,2,240,0,0,0,0,52,240,0,0,1,0],[1,240,0,0,0,240,0,0,0,0,52,189,0,0,1,189],[0,11,0,0,22,0,0,0,0,0,200,85,0,0,156,41] >;

C3×D206C4 in GAP, Magma, Sage, TeX

C_3\times D_{20}\rtimes_6C_4
% in TeX

G:=Group("C3xD20:6C4");
// GroupNames label

G:=SmallGroup(480,87);
// by ID

G=gap.SmallGroup(480,87);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,365,92,2524,1271,102,18822]);
// 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=b^-1,d*b*d^-1=b^11,d*c*d^-1=b^15*c>;
// generators/relations

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