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## G = Dic3×Dic5order 240 = 24·3·5

### Direct product of Dic3 and Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — Dic3×Dic5
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — Dic3×Dic5
 Lower central C15 — Dic3×Dic5
 Upper central C1 — C22

Generators and relations for Dic3×Dic5
G = < a,b,c,d | a6=c10=1, b2=a3, d2=c5, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 176 in 60 conjugacy classes, 36 normal (26 characteristic)
C1, C2 [×3], C3, C4 [×6], C22, C5, C6 [×3], C2×C4 [×3], C10 [×3], Dic3 [×2], Dic3 [×2], C12 [×2], C2×C6, C15, C42, Dic5 [×2], Dic5 [×2], C20 [×2], C2×C10, C2×Dic3, C2×Dic3, C2×C12, C30 [×3], C2×Dic5, C2×Dic5, C2×C20, C4×Dic3, C5×Dic3 [×2], C3×Dic5 [×2], Dic15 [×2], C2×C30, C4×Dic5, C6×Dic5, C10×Dic3, C2×Dic15, Dic3×Dic5
Quotients: C1, C2 [×3], C4 [×6], C22, S3, C2×C4 [×3], D5, Dic3 [×2], D6, C42, Dic5 [×2], D10, C4×S3 [×2], C2×Dic3, C4×D5 [×2], C2×Dic5, C4×Dic3, S3×D5, C4×Dic5, D5×Dic3, S3×Dic5, D30.C2, Dic3×Dic5

Smallest permutation representation of Dic3×Dic5
Regular action on 240 points
Generators in S240
(1 68 28 59 41 16)(2 69 29 60 42 17)(3 70 30 51 43 18)(4 61 21 52 44 19)(5 62 22 53 45 20)(6 63 23 54 46 11)(7 64 24 55 47 12)(8 65 25 56 48 13)(9 66 26 57 49 14)(10 67 27 58 50 15)(31 200 219 224 236 202)(32 191 220 225 237 203)(33 192 211 226 238 204)(34 193 212 227 239 205)(35 194 213 228 240 206)(36 195 214 229 231 207)(37 196 215 230 232 208)(38 197 216 221 233 209)(39 198 217 222 234 210)(40 199 218 223 235 201)(71 97 102 120 89 126)(72 98 103 111 90 127)(73 99 104 112 81 128)(74 100 105 113 82 129)(75 91 106 114 83 130)(76 92 107 115 84 121)(77 93 108 116 85 122)(78 94 109 117 86 123)(79 95 110 118 87 124)(80 96 101 119 88 125)(131 183 143 177 165 160)(132 184 144 178 166 151)(133 185 145 179 167 152)(134 186 146 180 168 153)(135 187 147 171 169 154)(136 188 148 172 170 155)(137 189 149 173 161 156)(138 190 150 174 162 157)(139 181 141 175 163 158)(140 182 142 176 164 159)
(1 114 59 75)(2 115 60 76)(3 116 51 77)(4 117 52 78)(5 118 53 79)(6 119 54 80)(7 120 55 71)(8 111 56 72)(9 112 57 73)(10 113 58 74)(11 88 23 96)(12 89 24 97)(13 90 25 98)(14 81 26 99)(15 82 27 100)(16 83 28 91)(17 84 29 92)(18 85 30 93)(19 86 21 94)(20 87 22 95)(31 187 224 169)(32 188 225 170)(33 189 226 161)(34 190 227 162)(35 181 228 163)(36 182 229 164)(37 183 230 165)(38 184 221 166)(39 185 222 167)(40 186 223 168)(41 130 68 106)(42 121 69 107)(43 122 70 108)(44 123 61 109)(45 124 62 110)(46 125 63 101)(47 126 64 102)(48 127 65 103)(49 128 66 104)(50 129 67 105)(131 232 177 196)(132 233 178 197)(133 234 179 198)(134 235 180 199)(135 236 171 200)(136 237 172 191)(137 238 173 192)(138 239 174 193)(139 240 175 194)(140 231 176 195)(141 213 158 206)(142 214 159 207)(143 215 160 208)(144 216 151 209)(145 217 152 210)(146 218 153 201)(147 219 154 202)(148 220 155 203)(149 211 156 204)(150 212 157 205)
(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)
(1 176 6 171)(2 175 7 180)(3 174 8 179)(4 173 9 178)(5 172 10 177)(11 147 16 142)(12 146 17 141)(13 145 18 150)(14 144 19 149)(15 143 20 148)(21 156 26 151)(22 155 27 160)(23 154 28 159)(24 153 29 158)(25 152 30 157)(31 106 36 101)(32 105 37 110)(33 104 38 109)(34 103 39 108)(35 102 40 107)(41 182 46 187)(42 181 47 186)(43 190 48 185)(44 189 49 184)(45 188 50 183)(51 138 56 133)(52 137 57 132)(53 136 58 131)(54 135 59 140)(55 134 60 139)(61 161 66 166)(62 170 67 165)(63 169 68 164)(64 168 69 163)(65 167 70 162)(71 235 76 240)(72 234 77 239)(73 233 78 238)(74 232 79 237)(75 231 80 236)(81 216 86 211)(82 215 87 220)(83 214 88 219)(84 213 89 218)(85 212 90 217)(91 207 96 202)(92 206 97 201)(93 205 98 210)(94 204 99 209)(95 203 100 208)(111 198 116 193)(112 197 117 192)(113 196 118 191)(114 195 119 200)(115 194 120 199)(121 228 126 223)(122 227 127 222)(123 226 128 221)(124 225 129 230)(125 224 130 229)

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

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

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

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + - + - + + - - + image C1 C2 C2 C2 C4 C4 C4 S3 D5 Dic3 D6 Dic5 D10 C4×S3 C4×D5 S3×D5 D5×Dic3 S3×Dic5 D30.C2 kernel Dic3×Dic5 C6×Dic5 C10×Dic3 C2×Dic15 C5×Dic3 C3×Dic5 Dic15 C2×Dic5 C2×Dic3 Dic5 C2×C10 Dic3 C2×C6 C10 C6 C22 C2 C2 C2 # reps 1 1 1 1 4 4 4 1 2 2 1 4 2 4 8 2 2 2 2

Matrix representation of Dic3×Dic5 in GL4(𝔽61) generated by

 1 0 0 0 0 1 0 0 0 0 60 15 0 0 12 2
,
 1 0 0 0 0 1 0 0 0 0 36 30 0 0 32 25
,
 0 60 0 0 1 18 0 0 0 0 60 0 0 0 0 60
,
 23 54 0 0 6 38 0 0 0 0 50 0 0 0 0 50
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,60,12,0,0,15,2],[1,0,0,0,0,1,0,0,0,0,36,32,0,0,30,25],[0,1,0,0,60,18,0,0,0,0,60,0,0,0,0,60],[23,6,0,0,54,38,0,0,0,0,50,0,0,0,0,50] >;

Dic3×Dic5 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times {\rm Dic}_5
% in TeX

G:=Group("Dic3xDic5");
// GroupNames label

G:=SmallGroup(240,25);
// by ID

G=gap.SmallGroup(240,25);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,55,490,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^6=c^10=1,b^2=a^3,d^2=c^5,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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