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G = D24⋊D5order 480 = 25·3·5

2nd semidirect product of D24 and D5 acting via D5/C5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D242D5, C4010D6, C242D10, D128D10, C12011C22, D10.13D12, C60.119C23, Dic5.15D12, D60.31C22, Dic3017C22, C82(S3×D5), C52C82D6, C6.4(D4×D5), (C5×D24)⋊4C2, C8⋊D53S3, C52(C8⋊D6), (C6×D5).2D4, (C4×D5).2D6, C30.4(C2×D4), C2.9(D5×D12), C24⋊D59C2, (D5×D12)⋊10C2, C31(D8⋊D5), C5⋊D2411C2, C152(C8⋊C22), C10.4(C2×D12), D125D58C2, (C3×Dic5).2D4, D12.D510C2, (C5×D12)⋊15C22, C20.67(C22×S3), (D5×C12).25C22, C12.142(C22×D5), C4.67(C2×S3×D5), (C3×C8⋊D5)⋊3C2, (C3×C52C8)⋊16C22, SmallGroup(480,326)

Series: Derived Chief Lower central Upper central

C1C60 — D24⋊D5
C1C5C15C30C60D5×C12D5×D12 — D24⋊D5
C15C30C60 — D24⋊D5
C1C2C4C8

Generators and relations for D24⋊D5
 G = < a,b,c,d | a24=b2=c5=d2=1, bab=a-1, ac=ca, dad=a13, bc=cb, dbd=a12b, dcd=c-1 >

Subgroups: 988 in 136 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×2], C22 [×6], C5, S3 [×3], C6, C6, C8, C8, C2×C4 [×2], D4 [×5], Q8, C23, D5 [×2], C10, C10 [×2], Dic3, C12, C12, D6 [×5], C2×C6, C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5, Dic5, C20, D10, D10 [×3], C2×C10 [×2], C24, C24, Dic6, C4×S3, D12 [×2], D12 [×2], C3⋊D4, C2×C12, C22×S3, C5×S3 [×2], C3×D5, D15, C30, C8⋊C22, C52C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4 [×2], C5×D4 [×2], C22×D5, C24⋊C2 [×2], D24, D24, C3×M4(2), C2×D12, C4○D12, C3×Dic5, Dic15, C60, S3×D5 [×2], C6×D5, S3×C10 [×2], D30, C8⋊D5, C40⋊C2, D4⋊D5, D4.D5, C5×D8, D4×D5, D42D5, C8⋊D6, C3×C52C8, C120, S3×Dic5, C15⋊D4, C5⋊D12, D5×C12, C5×D12 [×2], Dic30, D60, C2×S3×D5, D8⋊D5, C5⋊D24, D12.D5, C3×C8⋊D5, C5×D24, C24⋊D5, D125D5, D5×D12, D24⋊D5
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], D12 [×2], C22×S3, C8⋊C22, C22×D5, C2×D12, S3×D5, D4×D5, C8⋊D6, C2×S3×D5, D8⋊D5, D5×D12, D24⋊D5

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

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

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

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

51 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C5A5B6A6B8A8B10A10B10C10D10E10F12A12B12C15A15B20A20B24A24B24C24D30A30B40A40B40C40D60A60B60C60D120A···120H
order1222223444556688101010101010121212151520202424242430304040404060606060120···120
size11101212602210602222042022242424242220444444202044444444444···4

51 irreducible representations

dim111111112222222222244444444
type+++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D6D10D10D12D12C8⋊C22S3×D5D4×D5C8⋊D6C2×S3×D5D8⋊D5D5×D12D24⋊D5
kernelD24⋊D5C5⋊D24D12.D5C3×C8⋊D5C5×D24C24⋊D5D125D5D5×D12C8⋊D5C3×Dic5C6×D5D24C52C8C40C4×D5C24D12Dic5D10C15C8C6C5C4C3C2C1
# reps111111111112111242212222448

Matrix representation of D24⋊D5 in GL4(𝔽241) generated by

2204186174
372396755
55675730
174186211184
,
23937184211
20423057
1861742204
675537239
,
0100
24018900
0001
00240189
,
0100
1000
0001
0010
G:=sub<GL(4,GF(241))| [2,37,55,174,204,239,67,186,186,67,57,211,174,55,30,184],[239,204,186,67,37,2,174,55,184,30,2,37,211,57,204,239],[0,240,0,0,1,189,0,0,0,0,0,240,0,0,1,189],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

D24⋊D5 in GAP, Magma, Sage, TeX

D_{24}\rtimes D_5
% in TeX

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

G:=SmallGroup(480,326);
// by ID

G=gap.SmallGroup(480,326);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,422,135,142,346,80,1356,18822]);
// 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^13,b*c=c*b,d*b*d=a^12*b,d*c*d=c^-1>;
// generators/relations

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