direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D5×D24, C40⋊4D6, D12⋊7D10, C24⋊22D10, D120⋊11C2, C120⋊5C22, D10.23D12, D60⋊18C22, Dic5.6D12, C60.117C23, C3⋊1(D5×D8), C5⋊1(C2×D24), C15⋊1(C2×D8), (C3×D5)⋊1D8, (C8×D5)⋊1S3, C8⋊10(S3×D5), C6.2(D4×D5), (D5×D12)⋊8C2, (D5×C24)⋊1C2, (C5×D24)⋊2C2, C5⋊2C8⋊24D6, C5⋊D24⋊9C2, C30.2(C2×D4), C2.7(D5×D12), (C4×D5).77D6, (C6×D5).41D4, C10.2(C2×D12), (C5×D12)⋊14C22, C20.65(C22×S3), (C3×Dic5).45D4, (D5×C12).91C22, C12.140(C22×D5), C4.65(C2×S3×D5), (C3×C5⋊2C8)⋊28C22, SmallGroup(480,324)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D5×D24
G = < a,b,c,d | a5=b2=c24=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 1276 in 152 conjugacy classes, 44 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C8, C2×C4, D4, C23, D5, D5, C10, C10, C12, C12, D6, C2×C6, C15, C2×C8, D8, C2×D4, Dic5, C20, D10, D10, C2×C10, C24, C24, D12, D12, C2×C12, C22×S3, C5×S3, C3×D5, D15, C30, C2×D8, C5⋊2C8, C40, C4×D5, D20, C5⋊D4, C5×D4, C22×D5, D24, D24, C2×C24, C2×D12, C3×Dic5, C60, S3×D5, C6×D5, S3×C10, D30, C8×D5, D40, D4⋊D5, C5×D8, D4×D5, C2×D24, C3×C5⋊2C8, C120, C5⋊D12, D5×C12, C5×D12, D60, C2×S3×D5, D5×D8, C5⋊D24, D5×C24, C5×D24, D120, D5×D12, D5×D24
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, D8, C2×D4, D10, D12, C22×S3, C2×D8, C22×D5, D24, C2×D12, S3×D5, D4×D5, C2×D24, C2×S3×D5, D5×D8, D5×D12, D5×D24
►On 120 points
Generators in S120
(1 74 31 69 109)(2 75 32 70 110)(3 76 33 71 111)(4 77 34 72 112)(5 78 35 49 113)(6 79 36 50 114)(7 80 37 51 115)(8 81 38 52 116)(9 82 39 53 117)(10 83 40 54 118)(11 84 41 55 119)(12 85 42 56 120)(13 86 43 57 97)(14 87 44 58 98)(15 88 45 59 99)(16 89 46 60 100)(17 90 47 61 101)(18 91 48 62 102)(19 92 25 63 103)(20 93 26 64 104)(21 94 27 65 105)(22 95 28 66 106)(23 96 29 67 107)(24 73 30 68 108)
(1 97)(2 98)(3 99)(4 100)(5 101)(6 102)(7 103)(8 104)(9 105)(10 106)(11 107)(12 108)(13 109)(14 110)(15 111)(16 112)(17 113)(18 114)(19 115)(20 116)(21 117)(22 118)(23 119)(24 120)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(49 90)(50 91)(51 92)(52 93)(53 94)(54 95)(55 96)(56 73)(57 74)(58 75)(59 76)(60 77)(61 78)(62 79)(63 80)(64 81)(65 82)(66 83)(67 84)(68 85)(69 86)(70 87)(71 88)(72 89)
(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)
(1 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 24)(17 23)(18 22)(19 21)(25 27)(28 48)(29 47)(30 46)(31 45)(32 44)(33 43)(34 42)(35 41)(36 40)(37 39)(49 55)(50 54)(51 53)(56 72)(57 71)(58 70)(59 69)(60 68)(61 67)(62 66)(63 65)(73 89)(74 88)(75 87)(76 86)(77 85)(78 84)(79 83)(80 82)(90 96)(91 95)(92 94)(97 111)(98 110)(99 109)(100 108)(101 107)(102 106)(103 105)(112 120)(113 119)(114 118)(115 117)
G:=sub<Sym(120)| (1,74,31,69,109)(2,75,32,70,110)(3,76,33,71,111)(4,77,34,72,112)(5,78,35,49,113)(6,79,36,50,114)(7,80,37,51,115)(8,81,38,52,116)(9,82,39,53,117)(10,83,40,54,118)(11,84,41,55,119)(12,85,42,56,120)(13,86,43,57,97)(14,87,44,58,98)(15,88,45,59,99)(16,89,46,60,100)(17,90,47,61,101)(18,91,48,62,102)(19,92,25,63,103)(20,93,26,64,104)(21,94,27,65,105)(22,95,28,66,106)(23,96,29,67,107)(24,73,30,68,108), (1,97)(2,98)(3,99)(4,100)(5,101)(6,102)(7,103)(8,104)(9,105)(10,106)(11,107)(12,108)(13,109)(14,110)(15,111)(16,112)(17,113)(18,114)(19,115)(20,116)(21,117)(22,118)(23,119)(24,120)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(49,90)(50,91)(51,92)(52,93)(53,94)(54,95)(55,96)(56,73)(57,74)(58,75)(59,76)(60,77)(61,78)(62,79)(63,80)(64,81)(65,82)(66,83)(67,84)(68,85)(69,86)(70,87)(71,88)(72,89), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,24)(17,23)(18,22)(19,21)(25,27)(28,48)(29,47)(30,46)(31,45)(32,44)(33,43)(34,42)(35,41)(36,40)(37,39)(49,55)(50,54)(51,53)(56,72)(57,71)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65)(73,89)(74,88)(75,87)(76,86)(77,85)(78,84)(79,83)(80,82)(90,96)(91,95)(92,94)(97,111)(98,110)(99,109)(100,108)(101,107)(102,106)(103,105)(112,120)(113,119)(114,118)(115,117)>;
G:=Group( (1,74,31,69,109)(2,75,32,70,110)(3,76,33,71,111)(4,77,34,72,112)(5,78,35,49,113)(6,79,36,50,114)(7,80,37,51,115)(8,81,38,52,116)(9,82,39,53,117)(10,83,40,54,118)(11,84,41,55,119)(12,85,42,56,120)(13,86,43,57,97)(14,87,44,58,98)(15,88,45,59,99)(16,89,46,60,100)(17,90,47,61,101)(18,91,48,62,102)(19,92,25,63,103)(20,93,26,64,104)(21,94,27,65,105)(22,95,28,66,106)(23,96,29,67,107)(24,73,30,68,108), (1,97)(2,98)(3,99)(4,100)(5,101)(6,102)(7,103)(8,104)(9,105)(10,106)(11,107)(12,108)(13,109)(14,110)(15,111)(16,112)(17,113)(18,114)(19,115)(20,116)(21,117)(22,118)(23,119)(24,120)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(49,90)(50,91)(51,92)(52,93)(53,94)(54,95)(55,96)(56,73)(57,74)(58,75)(59,76)(60,77)(61,78)(62,79)(63,80)(64,81)(65,82)(66,83)(67,84)(68,85)(69,86)(70,87)(71,88)(72,89), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,24)(17,23)(18,22)(19,21)(25,27)(28,48)(29,47)(30,46)(31,45)(32,44)(33,43)(34,42)(35,41)(36,40)(37,39)(49,55)(50,54)(51,53)(56,72)(57,71)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65)(73,89)(74,88)(75,87)(76,86)(77,85)(78,84)(79,83)(80,82)(90,96)(91,95)(92,94)(97,111)(98,110)(99,109)(100,108)(101,107)(102,106)(103,105)(112,120)(113,119)(114,118)(115,117) );
G=PermutationGroup([[(1,74,31,69,109),(2,75,32,70,110),(3,76,33,71,111),(4,77,34,72,112),(5,78,35,49,113),(6,79,36,50,114),(7,80,37,51,115),(8,81,38,52,116),(9,82,39,53,117),(10,83,40,54,118),(11,84,41,55,119),(12,85,42,56,120),(13,86,43,57,97),(14,87,44,58,98),(15,88,45,59,99),(16,89,46,60,100),(17,90,47,61,101),(18,91,48,62,102),(19,92,25,63,103),(20,93,26,64,104),(21,94,27,65,105),(22,95,28,66,106),(23,96,29,67,107),(24,73,30,68,108)], [(1,97),(2,98),(3,99),(4,100),(5,101),(6,102),(7,103),(8,104),(9,105),(10,106),(11,107),(12,108),(13,109),(14,110),(15,111),(16,112),(17,113),(18,114),(19,115),(20,116),(21,117),(22,118),(23,119),(24,120),(25,37),(26,38),(27,39),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48),(49,90),(50,91),(51,92),(52,93),(53,94),(54,95),(55,96),(56,73),(57,74),(58,75),(59,76),(60,77),(61,78),(62,79),(63,80),(64,81),(65,82),(66,83),(67,84),(68,85),(69,86),(70,87),(71,88),(72,89)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,24),(17,23),(18,22),(19,21),(25,27),(28,48),(29,47),(30,46),(31,45),(32,44),(33,43),(34,42),(35,41),(36,40),(37,39),(49,55),(50,54),(51,53),(56,72),(57,71),(58,70),(59,69),(60,68),(61,67),(62,66),(63,65),(73,89),(74,88),(75,87),(76,86),(77,85),(78,84),(79,83),(80,82),(90,96),(91,95),(92,94),(97,111),(98,110),(99,109),(100,108),(101,107),(102,106),(103,105),(112,120),(113,119),(114,118),(115,117)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 24A | 24B | 24C | 24D | 24E | 24F | 24G | 24H | 30A | 30B | 40A | 40B | 40C | 40D | 60A | 60B | 60C | 60D | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 5 | 5 | 12 | 12 | 60 | 60 | 2 | 2 | 10 | 2 | 2 | 2 | 10 | 10 | 2 | 2 | 10 | 10 | 2 | 2 | 24 | 24 | 24 | 24 | 2 | 2 | 10 | 10 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | D8 | D10 | D10 | D12 | D12 | D24 | S3×D5 | D4×D5 | C2×S3×D5 | D5×D8 | D5×D12 | D5×D24 |
| kernel | D5×D24 | C5⋊D24 | D5×C24 | C5×D24 | D120 | D5×D12 | C8×D5 | C3×Dic5 | C6×D5 | D24 | C5⋊2C8 | C40 | C4×D5 | C3×D5 | C24 | D12 | Dic5 | D10 | D5 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 4 | 2 | 2 | 8 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of D5×D24 ►in GL4(𝔽241) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 240 | 189 |
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 232 | 105 | 0 | 0 |
| 136 | 127 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 240 | 0 | 0 |
| 240 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,0,240,0,0,1,189],[240,0,0,0,0,240,0,0,0,0,0,1,0,0,1,0],[232,136,0,0,105,127,0,0,0,0,1,0,0,0,0,1],[0,240,0,0,240,0,0,0,0,0,1,0,0,0,0,1] >;
D5×D24 in GAP, Magma, Sage, TeX
D_5\times D_{24} % in TeX
G:=Group("D5xD24"); // GroupNames label
G:=SmallGroup(480,324);
// by ID
G=gap.SmallGroup(480,324);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,135,142,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^2=c^24=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations