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G = D24.D5order 480 = 25·3·5

1st non-split extension by D24 of D5 acting via D5/C5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C40.8D6, C30.8D8, C155SD32, D24.1D5, C60.50D4, C20.2D12, C10.7D24, C24.41D10, Dic6010C2, C120.18C22, C52C162S3, C31(D8.D5), C8.17(S3×D5), C52(C48⋊C2), C6.2(D4⋊D5), (C5×D24).1C2, C2.5(C5⋊D24), C4.2(C5⋊D12), C12.52(C5⋊D4), (C3×C52C16)⋊2C2, SmallGroup(480,20)

Series: Derived Chief Lower central Upper central

C1C120 — D24.D5
C1C5C15C30C60C120C3×C52C16 — D24.D5
C15C30C60C120 — D24.D5
C1C2C4C8

Generators and relations for D24.D5
 G = < a,b,c,d | a24=b2=c5=1, d2=a9, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a21b, dcd-1=c-1 >

24C2
12C22
60C4
8S3
24C10
6D4
30Q8
4D6
20Dic3
12Dic5
12C2×C10
8C5×S3
3D8
5C16
15Q16
2D12
10Dic6
6C5×D4
6Dic10
4Dic15
4S3×C10
15SD32
5C48
5Dic12
3C5×D8
3Dic20
2Dic30
2C5×D12
5C48⋊C2
3D8.D5

Smallest permutation representation of D24.D5
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 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)(25 35)(26 34)(27 33)(28 32)(29 31)(36 48)(37 47)(38 46)(39 45)(40 44)(41 43)(49 63)(50 62)(51 61)(52 60)(53 59)(54 58)(55 57)(64 72)(65 71)(66 70)(67 69)(73 83)(74 82)(75 81)(76 80)(77 79)(84 96)(85 95)(86 94)(87 93)(88 92)(89 91)(97 104)(98 103)(99 102)(100 101)(105 120)(106 119)(107 118)(108 117)(109 116)(110 115)(111 114)(112 113)(121 138)(122 137)(123 136)(124 135)(125 134)(126 133)(127 132)(128 131)(129 130)(139 144)(140 143)(141 142)(145 155)(146 154)(147 153)(148 152)(149 151)(156 168)(157 167)(158 166)(159 165)(160 164)(161 163)(169 188)(170 187)(171 186)(172 185)(173 184)(174 183)(175 182)(176 181)(177 180)(178 179)(189 192)(190 191)(193 204)(194 203)(195 202)(196 201)(197 200)(198 199)(205 216)(206 215)(207 214)(208 213)(209 212)(210 211)(217 231)(218 230)(219 229)(220 228)(221 227)(222 226)(223 225)(232 240)(233 239)(234 238)(235 237)
(1 179 142 101 199)(2 180 143 102 200)(3 181 144 103 201)(4 182 121 104 202)(5 183 122 105 203)(6 184 123 106 204)(7 185 124 107 205)(8 186 125 108 206)(9 187 126 109 207)(10 188 127 110 208)(11 189 128 111 209)(12 190 129 112 210)(13 191 130 113 211)(14 192 131 114 212)(15 169 132 115 213)(16 170 133 116 214)(17 171 134 117 215)(18 172 135 118 216)(19 173 136 119 193)(20 174 137 120 194)(21 175 138 97 195)(22 176 139 98 196)(23 177 140 99 197)(24 178 141 100 198)(25 63 231 73 145)(26 64 232 74 146)(27 65 233 75 147)(28 66 234 76 148)(29 67 235 77 149)(30 68 236 78 150)(31 69 237 79 151)(32 70 238 80 152)(33 71 239 81 153)(34 72 240 82 154)(35 49 217 83 155)(36 50 218 84 156)(37 51 219 85 157)(38 52 220 86 158)(39 53 221 87 159)(40 54 222 88 160)(41 55 223 89 161)(42 56 224 90 162)(43 57 225 91 163)(44 58 226 92 164)(45 59 227 93 165)(46 60 228 94 166)(47 61 229 95 167)(48 62 230 96 168)
(1 55 10 64 19 49 4 58 13 67 22 52 7 61 16 70)(2 56 11 65 20 50 5 59 14 68 23 53 8 62 17 71)(3 57 12 66 21 51 6 60 15 69 24 54 9 63 18 72)(25 172 34 181 43 190 28 175 37 184 46 169 31 178 40 187)(26 173 35 182 44 191 29 176 38 185 47 170 32 179 41 188)(27 174 36 183 45 192 30 177 39 186 48 171 33 180 42 189)(73 118 82 103 91 112 76 97 85 106 94 115 79 100 88 109)(74 119 83 104 92 113 77 98 86 107 95 116 80 101 89 110)(75 120 84 105 93 114 78 99 87 108 96 117 81 102 90 111)(121 164 130 149 139 158 124 167 133 152 142 161 127 146 136 155)(122 165 131 150 140 159 125 168 134 153 143 162 128 147 137 156)(123 166 132 151 141 160 126 145 135 154 144 163 129 148 138 157)(193 217 202 226 211 235 196 220 205 229 214 238 199 223 208 232)(194 218 203 227 212 236 197 221 206 230 215 239 200 224 209 233)(195 219 204 228 213 237 198 222 207 231 216 240 201 225 210 234)

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

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

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

57 conjugacy classes

class 1 2A2B 3 4A4B5A5B 6 8A8B10A10B10C10D10E10F12A12B15A15B16A16B16C16D20A20B24A24B24C24D30A30B40A40B40C40D48A···48H60A60B60C60D120A···120H
order12234455688101010101010121215151616161620202424242430304040404048···4860606060120···120
size11242212022222222424242422441010101044222244444410···1044444···4

57 irreducible representations

dim111122222222222444444
type+++++++++++++++-+-
imageC1C2C2C2S3D4D5D6D8D10D12SD32C5⋊D4D24C48⋊C2S3×D5D4⋊D5C5⋊D12D8.D5C5⋊D24D24.D5
kernelD24.D5C3×C52C16C5×D24Dic60C52C16C60D24C40C30C24C20C15C12C10C5C8C6C4C3C2C1
# reps111111212224448222448

Matrix representation of D24.D5 in GL6(𝔽241)

691860000
1511940000
002395400
00174100
000010
000001
,
691860000
2181720000
00240000
00174100
000010
000001
,
100000
010000
001000
000100
000001
000024051
,
122360000
37840000
00240000
00024000
000022789
00009814

G:=sub<GL(6,GF(241))| [69,151,0,0,0,0,186,194,0,0,0,0,0,0,239,174,0,0,0,0,54,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[69,218,0,0,0,0,186,172,0,0,0,0,0,0,240,174,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,240,0,0,0,0,1,51],[122,37,0,0,0,0,36,84,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,227,98,0,0,0,0,89,14] >;

D24.D5 in GAP, Magma, Sage, TeX

D_{24}.D_5
% in TeX

G:=Group("D24.D5");
// GroupNames label

G:=SmallGroup(480,20);
// by ID

G=gap.SmallGroup(480,20);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,309,135,142,346,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^24=b^2=c^5=1,d^2=a^9,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^21*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Subgroup lattice of D24.D5 in TeX

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