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G = D5×D24order 480 = 25·3·5

Direct product of D5 and D24

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D5×D24, C404D6, D127D10, C2422D10, D12011C2, C1205C22, D10.23D12, D6018C22, Dic5.6D12, C60.117C23, C31(D5×D8), C51(C2×D24), C151(C2×D8), (C3×D5)⋊1D8, (C8×D5)⋊1S3, C810(S3×D5), C6.2(D4×D5), (D5×D12)⋊8C2, (D5×C24)⋊1C2, (C5×D24)⋊2C2, C52C824D6, C5⋊D249C2, 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×C52C8)⋊28C22, SmallGroup(480,324)

Series: Derived Chief Lower central Upper central

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

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 2A2B2C2D2E2F2G 3 4A4B5A5B6A6B6C8A8B8C8D10A10B10C10D10E10F12A12B12C12D15A15B20A20B24A24B24C24D24E24F24G24H30A30B40A40B40C40D60A60B60C60D120A···120H
order122222223445566688881010101010101212121215152020242424242424242430304040404060606060120···120
size115512126060221022210102210102224242424221010444422221010101044444444444···4

60 irreducible representations

dim1111112222222222222444444
type+++++++++++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D5D6D6D6D8D10D10D12D12D24S3×D5D4×D5C2×S3×D5D5×D8D5×D12D5×D24
kernelD5×D24C5⋊D24D5×C24C5×D24D120D5×D12C8×D5C3×Dic5C6×D5D24C52C8C40C4×D5C3×D5C24D12Dic5D10D5C8C6C4C3C2C1
# reps1211121112111424228222448

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

1000
0100
0001
00240189
,
240000
024000
0001
0010
,
23210500
13612700
0010
0001
,
024000
240000
0010
0001
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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