D24:5D5 - GroupNames

G = D₂₄⋊₅D₅ order 480 = 2⁵·3·5

5^th semidirect product of D₂₄ and D₅ acting via D₅/C₅=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₂₄⋊₅D₅, C₄₀.₂₁D₆, Dic₂₀⋊₅S₃, D₃₀.₂₅D₄, D₁₂.₄D₁₀, C₂₄.₂₁D₁₀, Dic₁₀.₄D₆, C₆₀.₁₄₇C₂³, C₁₂₀.₃₈C₂², Dic₁₅.₃₀D₄, (C₅×D₂₄)⋊₇C₂, (C₈×D₁₅)⋊₈C₂, C₁₅⋊₉(C₄○D₈), C₈.₃₁(S₃×D₅), C₆.₃₆(D₄×D₅), C₃⋊₂(D₈⋊₃D₅), C₁₀.₃₆(S₃×D₄), C₃₀.₂₉(C₂×D₄), C₅⋊₂(D₂₄⋊C₂), (C₃×Dic₂₀)⋊₈C₂, D₁₂⋊D₅⋊₁₀C₂, C₂₀.D₆⋊₁₁C₂, C₂₀.₈₂(C₂²×S₃), C₁₂.₈₂(C₂²×D₅), C₂.₁₄(C₂₀⋊D₆), C₁₅⋊₃C₈.₄₃C₂², (C₅×D₁₂).₂₈C₂², (C₄×D₁₅).₅₈C₂², (C₃×Dic₁₀).₃₀C₂², C₄.₁₂₀(C₂×S₃×D₅), SmallGroup(480,355)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆₀ — D₂₄⋊₅D₅

Chief series

C₁ — C₅ — C₁₅ — C₃₀ — C₆₀ — C₃×Dic₁₀ — D₁₂⋊D₅ — D₂₄⋊₅D₅

Lower central

C₁₅ — C₃₀ — C₆₀ — D₂₄⋊₅D₅

Upper central

C₁ — C₂ — C₄ — C₈

Generators and relations for D₂₄⋊₅D₅
G = < a,b,c,d | a²⁴=b²=c⁵=d²=1, bab=a^-1, ac=ca, dad=a¹⁷, bc=cb, dbd=a⁴b, dcd=c^-1 >

Subgroups: 700 in 124 conjugacy classes, 38 normal (26 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₅, S₃, C₆, C₈, C₈, C₂×C₄, D₄, Q₈, D₅, C₁₀, C₁₀, Dic₃, C₁₂, C₁₂, D₆, C₁₅, C₂×C₈, D₈, SD₁₆, Q₁₆, C₄○D₄, Dic₅, C₂₀, D₁₀, C₂×C₁₀, C₃⋊C₈, C₂₄, C₄×S₃, D₁₂, D₁₂, C₃×Q₈, C₅×S₃, D₁₅, C₃₀, C₄○D₈, C₅⋊₂C₈, C₄₀, Dic₁₀, C₄×D₅, C₂×Dic₅, C₅⋊D₄, C₅×D₄, S₃×C₈, D₂₄, Q₈⋊₂S₃, C₃×Q₁₆, Q₈⋊₃S₃, C₃×Dic₅, Dic₁₅, C₆₀, S₃×C₁₀, D₃₀, C₈×D₅, Dic₂₀, D₄.D₅, C₅×D₈, D₄⋊₂D₅, D₂₄⋊C₂, C₁₅⋊₃C₈, C₁₂₀, S₃×Dic₅, C₅⋊D₁₂, C₃×Dic₁₀, C₅×D₁₂, C₄×D₁₅, D₈⋊₃D₅, C₂₀.D₆, C₃×Dic₂₀, C₅×D₂₄, C₈×D₁₅, D₁₂⋊D₅, D₂₄⋊₅D₅
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₅, D₆, C₂×D₄, D₁₀, C₂²×S₃, C₄○D₈, C₂²×D₅, S₃×D₄, S₃×D₅, D₄×D₅, D₂₄⋊C₂, C₂×S₃×D₅, D₈⋊₃D₅, C₂₀⋊D₆, D₂₄⋊₅D₅

Smallest permutation representation of D₂₄⋊₅D₅
►On 240 points

Generators in S₂₄₀

(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)

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

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

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

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

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

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

51 conjugacy classes

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

51 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+	+	+		+	+	+	+	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	S₃	D₄	D₄	D₅	D₆	D₆	D₁₀	D₁₀	C₄○D₈	S₃×D₄	S₃×D₅	D₄×D₅	D₂₄⋊C₂	C₂×S₃×D₅	D₈⋊₃D₅	C₂₀⋊D₆	D₂₄⋊₅D₅
kernel	D₂₄⋊₅D₅	C₂₀.D₆	C₃×Dic₂₀	C₅×D₂₄	C₈×D₁₅	D₁₂⋊D₅	Dic₂₀	Dic₁₅	D₃₀	D₂₄	C₄₀	Dic₁₀	C₂₄	D₁₂	C₁₅	C₁₀	C₈	C₆	C₅	C₄	C₃	C₂	C₁
# reps	1	2	1	1	1	2	1	1	1	2	1	2	2	4	4	1	2	2	2	2	4	4	8

Matrix representation of D₂₄⋊₅D₅ ►in GL₆(𝔽₂₄₁)

0	240	0	0	0	0
1	1	0	0	0	0
0	0	240	0	0	0
0	0	0	240	0	0
0	0	0	0	8	0
0	0	0	0	105	211

0	240	0	0	0	0
240	0	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	97	59
0	0	0	0	200	144

1	0	0	0	0	0
0	1	0	0	0	0
0	0	240	1	0	0
0	0	50	190	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	0	0	0	0	0
240	240	0	0	0	0
0	0	240	0	0	0
0	0	50	1	0	0
0	0	0	0	1	0
0	0	0	0	107	240

G:=sub<GL(6,GF(241))| [0,1,0,0,0,0,240,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,8,105,0,0,0,0,0,211],[0,240,0,0,0,0,240,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,97,200,0,0,0,0,59,144],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,50,0,0,0,0,1,190,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,240,0,0,0,0,0,240,0,0,0,0,0,0,240,50,0,0,0,0,0,1,0,0,0,0,0,0,1,107,0,0,0,0,0,240] >;

D₂₄⋊₅D₅ in GAP, Magma, Sage, TeX

D_{24}\rtimes_5D_5

% in TeX

G:=Group("D24:5D5");

// GroupNames label

G:=SmallGroup(480,355);

// by ID

G=gap.SmallGroup(480,355);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,254,135,142,675,346,80,1356,18822]);

// Polycyclic

G:=Group<a,b,c,d|a^24=b^2=c^5=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^17,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;

// generators/relations

G = D24⋊5D5 order 480 = 25·3·5

5th semidirect product of D24 and D5 acting via D5/C5=C2

G = D₂₄⋊₅D₅ order 480 = 2⁵·3·5

5^th semidirect product of D₂₄ and D₅ acting via D₅/C₅=C₂