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## G = D5×D24order 480 = 25·3·5

### Direct product of D5 and D24

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — D5×D24
 Chief series C1 — C5 — C15 — C30 — C60 — D5×C12 — D5×D12 — D5×D24
 Lower central C15 — C30 — C60 — D5×D24
 Upper central C1 — C2 — C4 — C8

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 [×6], C3, C4, C4, C22 [×9], C5, S3 [×4], C6, C6 [×2], C8, C8, C2×C4, D4 [×6], C23 [×2], D5 [×2], D5 [×2], C10, C10 [×2], C12, C12, D6 [×8], C2×C6, C15, C2×C8, D8 [×4], C2×D4 [×2], Dic5, C20, D10, D10 [×6], C2×C10 [×2], C24, C24, D12 [×2], D12 [×4], C2×C12, C22×S3 [×2], C5×S3 [×2], C3×D5 [×2], D15 [×2], C30, C2×D8, C52C8, C40, C4×D5, D20 [×2], C5⋊D4 [×2], C5×D4 [×2], C22×D5 [×2], D24, D24 [×3], C2×C24, C2×D12 [×2], C3×Dic5, C60, S3×D5 [×4], C6×D5, S3×C10 [×2], D30 [×2], C8×D5, D40, D4⋊D5 [×2], C5×D8, D4×D5 [×2], C2×D24, C3×C52C8, C120, C5⋊D12 [×2], D5×C12, C5×D12 [×2], D60 [×2], C2×S3×D5 [×2], D5×D8, C5⋊D24 [×2], D5×C24, C5×D24, D120, D5×D12 [×2], D5×D24
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], D8 [×2], C2×D4, D10 [×3], D12 [×2], C22×S3, C2×D8, C22×D5, D24 [×2], C2×D12, S3×D5, D4×D5, C2×D24, C2×S3×D5, D5×D8, D5×D12, D5×D24

Smallest permutation representation of D5×D24
On 120 points
Generators in S120
(1 33 86 104 60)(2 34 87 105 61)(3 35 88 106 62)(4 36 89 107 63)(5 37 90 108 64)(6 38 91 109 65)(7 39 92 110 66)(8 40 93 111 67)(9 41 94 112 68)(10 42 95 113 69)(11 43 96 114 70)(12 44 73 115 71)(13 45 74 116 72)(14 46 75 117 49)(15 47 76 118 50)(16 48 77 119 51)(17 25 78 120 52)(18 26 79 97 53)(19 27 80 98 54)(20 28 81 99 55)(21 29 82 100 56)(22 30 83 101 57)(23 31 84 102 58)(24 32 85 103 59)
(1 72)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 55)(9 56)(10 57)(11 58)(12 59)(13 60)(14 61)(15 62)(16 63)(17 64)(18 65)(19 66)(20 67)(21 68)(22 69)(23 70)(24 71)(25 108)(26 109)(27 110)(28 111)(29 112)(30 113)(31 114)(32 115)(33 116)(34 117)(35 118)(36 119)(37 120)(38 97)(39 98)(40 99)(41 100)(42 101)(43 102)(44 103)(45 104)(46 105)(47 106)(48 107)(73 85)(74 86)(75 87)(76 88)(77 89)(78 90)(79 91)(80 92)(81 93)(82 94)(83 95)(84 96)
(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 31)(26 30)(27 29)(32 48)(33 47)(34 46)(35 45)(36 44)(37 43)(38 42)(39 41)(49 61)(50 60)(51 59)(52 58)(53 57)(54 56)(62 72)(63 71)(64 70)(65 69)(66 68)(73 89)(74 88)(75 87)(76 86)(77 85)(78 84)(79 83)(80 82)(90 96)(91 95)(92 94)(97 101)(98 100)(102 120)(103 119)(104 118)(105 117)(106 116)(107 115)(108 114)(109 113)(110 112)

G:=sub<Sym(120)| (1,33,86,104,60)(2,34,87,105,61)(3,35,88,106,62)(4,36,89,107,63)(5,37,90,108,64)(6,38,91,109,65)(7,39,92,110,66)(8,40,93,111,67)(9,41,94,112,68)(10,42,95,113,69)(11,43,96,114,70)(12,44,73,115,71)(13,45,74,116,72)(14,46,75,117,49)(15,47,76,118,50)(16,48,77,119,51)(17,25,78,120,52)(18,26,79,97,53)(19,27,80,98,54)(20,28,81,99,55)(21,29,82,100,56)(22,30,83,101,57)(23,31,84,102,58)(24,32,85,103,59), (1,72)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,56)(10,57)(11,58)(12,59)(13,60)(14,61)(15,62)(16,63)(17,64)(18,65)(19,66)(20,67)(21,68)(22,69)(23,70)(24,71)(25,108)(26,109)(27,110)(28,111)(29,112)(30,113)(31,114)(32,115)(33,116)(34,117)(35,118)(36,119)(37,120)(38,97)(39,98)(40,99)(41,100)(42,101)(43,102)(44,103)(45,104)(46,105)(47,106)(48,107)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96), (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)(25,31)(26,30)(27,29)(32,48)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(49,61)(50,60)(51,59)(52,58)(53,57)(54,56)(62,72)(63,71)(64,70)(65,69)(66,68)(73,89)(74,88)(75,87)(76,86)(77,85)(78,84)(79,83)(80,82)(90,96)(91,95)(92,94)(97,101)(98,100)(102,120)(103,119)(104,118)(105,117)(106,116)(107,115)(108,114)(109,113)(110,112)>;

G:=Group( (1,33,86,104,60)(2,34,87,105,61)(3,35,88,106,62)(4,36,89,107,63)(5,37,90,108,64)(6,38,91,109,65)(7,39,92,110,66)(8,40,93,111,67)(9,41,94,112,68)(10,42,95,113,69)(11,43,96,114,70)(12,44,73,115,71)(13,45,74,116,72)(14,46,75,117,49)(15,47,76,118,50)(16,48,77,119,51)(17,25,78,120,52)(18,26,79,97,53)(19,27,80,98,54)(20,28,81,99,55)(21,29,82,100,56)(22,30,83,101,57)(23,31,84,102,58)(24,32,85,103,59), (1,72)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,56)(10,57)(11,58)(12,59)(13,60)(14,61)(15,62)(16,63)(17,64)(18,65)(19,66)(20,67)(21,68)(22,69)(23,70)(24,71)(25,108)(26,109)(27,110)(28,111)(29,112)(30,113)(31,114)(32,115)(33,116)(34,117)(35,118)(36,119)(37,120)(38,97)(39,98)(40,99)(41,100)(42,101)(43,102)(44,103)(45,104)(46,105)(47,106)(48,107)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96), (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)(25,31)(26,30)(27,29)(32,48)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(49,61)(50,60)(51,59)(52,58)(53,57)(54,56)(62,72)(63,71)(64,70)(65,69)(66,68)(73,89)(74,88)(75,87)(76,86)(77,85)(78,84)(79,83)(80,82)(90,96)(91,95)(92,94)(97,101)(98,100)(102,120)(103,119)(104,118)(105,117)(106,116)(107,115)(108,114)(109,113)(110,112) );

G=PermutationGroup([(1,33,86,104,60),(2,34,87,105,61),(3,35,88,106,62),(4,36,89,107,63),(5,37,90,108,64),(6,38,91,109,65),(7,39,92,110,66),(8,40,93,111,67),(9,41,94,112,68),(10,42,95,113,69),(11,43,96,114,70),(12,44,73,115,71),(13,45,74,116,72),(14,46,75,117,49),(15,47,76,118,50),(16,48,77,119,51),(17,25,78,120,52),(18,26,79,97,53),(19,27,80,98,54),(20,28,81,99,55),(21,29,82,100,56),(22,30,83,101,57),(23,31,84,102,58),(24,32,85,103,59)], [(1,72),(2,49),(3,50),(4,51),(5,52),(6,53),(7,54),(8,55),(9,56),(10,57),(11,58),(12,59),(13,60),(14,61),(15,62),(16,63),(17,64),(18,65),(19,66),(20,67),(21,68),(22,69),(23,70),(24,71),(25,108),(26,109),(27,110),(28,111),(29,112),(30,113),(31,114),(32,115),(33,116),(34,117),(35,118),(36,119),(37,120),(38,97),(39,98),(40,99),(41,100),(42,101),(43,102),(44,103),(45,104),(46,105),(47,106),(48,107),(73,85),(74,86),(75,87),(76,88),(77,89),(78,90),(79,91),(80,92),(81,93),(82,94),(83,95),(84,96)], [(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),(25,31),(26,30),(27,29),(32,48),(33,47),(34,46),(35,45),(36,44),(37,43),(38,42),(39,41),(49,61),(50,60),(51,59),(52,58),(53,57),(54,56),(62,72),(63,71),(64,70),(65,69),(66,68),(73,89),(74,88),(75,87),(76,86),(77,85),(78,84),(79,83),(80,82),(90,96),(91,95),(92,94),(97,101),(98,100),(102,120),(103,119),(104,118),(105,117),(106,116),(107,115),(108,114),(109,113),(110,112)])

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

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