Copied to
clipboard

G = D5xD24order 480 = 25·3·5

Direct product of D5 and D24

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

Aliases: D5xD24, C40:4D6, D12:7D10, C24:22D10, D120:11C2, C120:5C22, D10.23D12, D60:18C22, Dic5.6D12, C60.117C23, C3:1(D5xD8), C5:1(C2xD24), C15:1(C2xD8), (C3xD5):1D8, (C8xD5):1S3, C8:10(S3xD5), C6.2(D4xD5), (D5xD12):8C2, (D5xC24):1C2, (C5xD24):2C2, C5:2C8:24D6, C5:D24:9C2, C30.2(C2xD4), C2.7(D5xD12), (C4xD5).77D6, (C6xD5).41D4, C10.2(C2xD12), (C5xD12):14C22, C20.65(C22xS3), (C3xDic5).45D4, (D5xC12).91C22, C12.140(C22xD5), C4.65(C2xS3xD5), (C3xC5:2C8):28C22, SmallGroup(480,324)

Series: Derived Chief Lower central Upper central

C1C60 — D5xD24
C1C5C15C30C60D5xC12D5xD12 — D5xD24
C15C30C60 — D5xD24
C1C2C4C8

Generators and relations for D5xD24
 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, C2xC4, D4, C23, D5, D5, C10, C10, C12, C12, D6, C2xC6, C15, C2xC8, D8, C2xD4, Dic5, C20, D10, D10, C2xC10, C24, C24, D12, D12, C2xC12, C22xS3, C5xS3, C3xD5, D15, C30, C2xD8, C5:2C8, C40, C4xD5, D20, C5:D4, C5xD4, C22xD5, D24, D24, C2xC24, C2xD12, C3xDic5, C60, S3xD5, C6xD5, S3xC10, D30, C8xD5, D40, D4:D5, C5xD8, D4xD5, C2xD24, C3xC5:2C8, C120, C5:D12, D5xC12, C5xD12, D60, C2xS3xD5, D5xD8, C5:D24, D5xC24, C5xD24, D120, D5xD12, D5xD24
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, D8, C2xD4, D10, D12, C22xS3, C2xD8, C22xD5, D24, C2xD12, S3xD5, D4xD5, C2xD24, C2xS3xD5, D5xD8, D5xD12, D5xD24

Smallest permutation representation of D5xD24
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 2A2B2C2D2E2F2G 3 4A4B5A5B6A6B6C8A8B8C8D10A10B10C10D10E10F12A12B12C12D15A15B20A20B24A24B24C24D24E24F24G24H30A30B40A40B40C40D60A60B60C60D120A···120H
order122222223445566688881010101010101212121215152020242424242424242430304040404060606060120···120
size115512126060221022210102210102224242424221010444422221010101044444444444···4

60 irreducible representations

dim1111112222222222222444444
type+++++++++++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D5D6D6D6D8D10D10D12D12D24S3xD5D4xD5C2xS3xD5D5xD8D5xD12D5xD24
kernelD5xD24C5:D24D5xC24C5xD24D120D5xD12C8xD5C3xDic5C6xD5D24C5:2C8C40C4xD5C3xD5C24D12Dic5D10D5C8C6C4C3C2C1
# reps1211121112111424228222448

Matrix representation of D5xD24 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] >;

D5xD24 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

׿
x
:
Z
F
o
wr
Q
<