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## G = C3×C20.55D4order 480 = 25·3·5

### Direct product of C3 and C20.55D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×C20.55D4
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×C60 — C6×C5⋊2C8 — C3×C20.55D4
 Lower central C5 — C10 — C3×C20.55D4
 Upper central C1 — C2×C12 — C22×C12

Generators and relations for C3×C20.55D4
G = < a,b,c,d | a3=b20=1, c4=b10, d2=b5, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b9, dcd-1=b15c3 >

Subgroups: 192 in 100 conjugacy classes, 58 normal (38 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, C23, C10, C10, C12, C12, C2×C6, C2×C6, C2×C6, C15, C2×C8, C22×C4, C20, C20, C2×C10, C2×C10, C2×C10, C24, C2×C12, C2×C12, C22×C6, C30, C30, C22⋊C8, C52C8, C2×C20, C2×C20, C22×C10, C2×C24, C22×C12, C60, C60, C2×C30, C2×C30, C2×C30, C2×C52C8, C22×C20, C3×C22⋊C8, C3×C52C8, C2×C60, C2×C60, C22×C30, C20.55D4, C6×C52C8, C22×C60, C3×C20.55D4
Quotients:

Smallest permutation representation of C3×C20.55D4
On 240 points
Generators in S240
(1 123 145)(2 124 146)(3 125 147)(4 126 148)(5 127 149)(6 128 150)(7 129 151)(8 130 152)(9 131 153)(10 132 154)(11 133 155)(12 134 156)(13 135 157)(14 136 158)(15 137 159)(16 138 160)(17 139 141)(18 140 142)(19 121 143)(20 122 144)(21 120 44)(22 101 45)(23 102 46)(24 103 47)(25 104 48)(26 105 49)(27 106 50)(28 107 51)(29 108 52)(30 109 53)(31 110 54)(32 111 55)(33 112 56)(34 113 57)(35 114 58)(36 115 59)(37 116 60)(38 117 41)(39 118 42)(40 119 43)(61 232 168)(62 233 169)(63 234 170)(64 235 171)(65 236 172)(66 237 173)(67 238 174)(68 239 175)(69 240 176)(70 221 177)(71 222 178)(72 223 179)(73 224 180)(74 225 161)(75 226 162)(76 227 163)(77 228 164)(78 229 165)(79 230 166)(80 231 167)(81 212 192)(82 213 193)(83 214 194)(84 215 195)(85 216 196)(86 217 197)(87 218 198)(88 219 199)(89 220 200)(90 201 181)(91 202 182)(92 203 183)(93 204 184)(94 205 185)(95 206 186)(96 207 187)(97 208 188)(98 209 189)(99 210 190)(100 211 191)
(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 41 100 79 11 51 90 69)(2 50 81 68 12 60 91 78)(3 59 82 77 13 49 92 67)(4 48 83 66 14 58 93 76)(5 57 84 75 15 47 94 65)(6 46 85 64 16 56 95 74)(7 55 86 73 17 45 96 63)(8 44 87 62 18 54 97 72)(9 53 88 71 19 43 98 61)(10 42 89 80 20 52 99 70)(21 218 233 140 31 208 223 130)(22 207 234 129 32 217 224 139)(23 216 235 138 33 206 225 128)(24 205 236 127 34 215 226 137)(25 214 237 136 35 204 227 126)(26 203 238 125 36 213 228 135)(27 212 239 134 37 202 229 124)(28 201 240 123 38 211 230 133)(29 210 221 132 39 220 231 122)(30 219 222 121 40 209 232 131)(101 187 170 151 111 197 180 141)(102 196 171 160 112 186 161 150)(103 185 172 149 113 195 162 159)(104 194 173 158 114 184 163 148)(105 183 174 147 115 193 164 157)(106 192 175 156 116 182 165 146)(107 181 176 145 117 191 166 155)(108 190 177 154 118 200 167 144)(109 199 178 143 119 189 168 153)(110 188 179 152 120 198 169 142)
(1 74 6 79 11 64 16 69)(2 63 7 68 12 73 17 78)(3 72 8 77 13 62 18 67)(4 61 9 66 14 71 19 76)(5 70 10 75 15 80 20 65)(21 218 26 203 31 208 36 213)(22 207 27 212 32 217 37 202)(23 216 28 201 33 206 38 211)(24 205 29 210 34 215 39 220)(25 214 30 219 35 204 40 209)(41 100 46 85 51 90 56 95)(42 89 47 94 52 99 57 84)(43 98 48 83 53 88 58 93)(44 87 49 92 54 97 59 82)(45 96 50 81 55 86 60 91)(101 187 106 192 111 197 116 182)(102 196 107 181 112 186 117 191)(103 185 108 190 113 195 118 200)(104 194 109 199 114 184 119 189)(105 183 110 188 115 193 120 198)(121 227 126 232 131 237 136 222)(122 236 127 221 132 226 137 231)(123 225 128 230 133 235 138 240)(124 234 129 239 134 224 139 229)(125 223 130 228 135 233 140 238)(141 165 146 170 151 175 156 180)(142 174 147 179 152 164 157 169)(143 163 148 168 153 173 158 178)(144 172 149 177 154 162 159 167)(145 161 150 166 155 171 160 176)

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

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

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

156 conjugacy classes

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

156 irreducible representations

 dim 1 1 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 type + + + + + - + - image C1 C2 C2 C3 C4 C4 C6 C6 C8 C12 C12 C24 D4 D5 M4(2) Dic5 D10 Dic5 C3×D4 C3×D5 C5⋊D4 C5⋊2C8 C3×M4(2) C3×Dic5 C6×D5 C3×Dic5 C4.Dic5 C3×C5⋊D4 C3×C5⋊2C8 C3×C4.Dic5 kernel C3×C20.55D4 C6×C5⋊2C8 C22×C60 C20.55D4 C2×C60 C22×C30 C2×C5⋊2C8 C22×C20 C2×C30 C2×C20 C22×C10 C2×C10 C60 C22×C12 C30 C2×C12 C2×C12 C22×C6 C20 C22×C4 C12 C2×C6 C10 C2×C4 C2×C4 C23 C6 C4 C22 C2 # reps 1 2 1 2 2 2 4 2 8 4 4 16 2 2 2 2 2 2 4 4 8 8 4 4 4 4 8 16 16 16

Matrix representation of C3×C20.55D4 in GL3(𝔽241) generated by

 1 0 0 0 225 0 0 0 225
,
 177 0 0 0 25 0 0 0 106
,
 30 0 0 0 0 1 0 177 0
,
 211 0 0 0 0 1 0 64 0
G:=sub<GL(3,GF(241))| [1,0,0,0,225,0,0,0,225],[177,0,0,0,25,0,0,0,106],[30,0,0,0,0,177,0,1,0],[211,0,0,0,0,64,0,1,0] >;

C3×C20.55D4 in GAP, Magma, Sage, TeX

C_3\times C_{20}._{55}D_4
% in TeX

G:=Group("C3xC20.55D4");
// GroupNames label

G:=SmallGroup(480,108);
// by ID

G=gap.SmallGroup(480,108);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,365,136,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^20=1,c^4=b^10,d^2=b^5,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^9,d*c*d^-1=b^15*c^3>;
// generators/relations

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