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G = D5×C24order 240 = 24·3·5

Direct product of C24 and D5

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D5×C24, C403C6, C1209C2, D10.4C12, C12.56D10, C60.69C22, Dic5.4C12, C53(C2×C24), C52C86C6, C1511(C2×C8), (C6×D5).8C4, (C4×D5).7C6, C2.1(D5×C12), C4.12(C6×D5), C6.15(C4×D5), C30.40(C2×C4), C10.8(C2×C12), C20.13(C2×C6), (D5×C12).14C2, (C3×Dic5).8C4, (C3×C52C8)⋊13C2, SmallGroup(240,33)

Series: Derived Chief Lower central Upper central

C1C5 — D5×C24
C1C5C10C20C60D5×C12 — D5×C24
C5 — D5×C24
C1C24

Generators and relations for D5×C24
 G = < a,b,c | a24=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

5C2
5C2
5C22
5C4
5C6
5C6
5C8
5C2×C4
5C12
5C2×C6
5C2×C8
5C24
5C2×C12
5C2×C24

Smallest permutation representation of D5×C24
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 64 28 96 112)(2 65 29 73 113)(3 66 30 74 114)(4 67 31 75 115)(5 68 32 76 116)(6 69 33 77 117)(7 70 34 78 118)(8 71 35 79 119)(9 72 36 80 120)(10 49 37 81 97)(11 50 38 82 98)(12 51 39 83 99)(13 52 40 84 100)(14 53 41 85 101)(15 54 42 86 102)(16 55 43 87 103)(17 56 44 88 104)(18 57 45 89 105)(19 58 46 90 106)(20 59 47 91 107)(21 60 48 92 108)(22 61 25 93 109)(23 62 26 94 110)(24 63 27 95 111)
(1 100)(2 101)(3 102)(4 103)(5 104)(6 105)(7 106)(8 107)(9 108)(10 109)(11 110)(12 111)(13 112)(14 113)(15 114)(16 115)(17 116)(18 117)(19 118)(20 119)(21 120)(22 97)(23 98)(24 99)(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 93)(50 94)(51 95)(52 96)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(61 81)(62 82)(63 83)(64 84)(65 85)(66 86)(67 87)(68 88)(69 89)(70 90)(71 91)(72 92)

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

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

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

D5×C24 is a maximal subgroup of
C40.51D6  C24.F5  C120.C4  C24⋊F5  C120⋊C4  D5.D24  C40.Dic3  C24.1F5  C40.54D6  C40.34D6  C40.31D6  D247D5  D120⋊C2

96 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D5A5B6A6B6C6D6E6F8A8B8C8D8E8F8G8H10A10B12A12B12C12D12E12F12G12H15A15B15C15D20A20B20C20D24A···24H24I···24P30A30B30C30D40A···40H60A···60H120A···120P
order1222334444556666668888888810101212121212121212151515152020202024···2424···243030303040···4060···60120···120
size115511115522115555111155552211115555222222221···15···522222···22···22···2

96 irreducible representations

dim1111111111111122222222
type++++++
imageC1C2C2C2C3C4C4C6C6C6C8C12C12C24D5D10C3×D5C4×D5C6×D5C8×D5D5×C12D5×C24
kernelD5×C24C3×C52C8C120D5×C12C8×D5C3×Dic5C6×D5C52C8C40C4×D5C3×D5Dic5D10D5C24C12C8C6C4C3C2C1
# reps111122222284416224448816

Matrix representation of D5×C24 in GL2(𝔽241) generated by

1210
0121
,
2401
50190
,
10
191240
G:=sub<GL(2,GF(241))| [121,0,0,121],[240,50,1,190],[1,191,0,240] >;

D5×C24 in GAP, Magma, Sage, TeX

D_5\times C_{24}
% in TeX

G:=Group("D5xC24");
// GroupNames label

G:=SmallGroup(240,33);
// by ID

G=gap.SmallGroup(240,33);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-5,79,69,6917]);
// Polycyclic

G:=Group<a,b,c|a^24=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of D5×C24 in TeX

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