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G = D120  order 240 = 24·3·5

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D120, C5⋊1D24, C3⋊1D40, C15⋊4D8, C40⋊1S3, C24⋊1D5, C8⋊1D15, C120⋊1C2, D60⋊1C2, C6.2D20, C2.4D60, C4.9D30, C10.2D12, C20.44D6, C30.20D4, C12.44D10, C60.51C22, sometimes denoted D240 or Dih120 or Dih240, SmallGroup(240,68)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C60 — D120
Chief series
C1 — C5 — C15 — C30 — C60 — D60 — D120
Lower central
C15 — C30 — C60 — D120
Upper central
C1 — C2 — C4 — C8

Generators and relations for D120
 G = < a,b | a120=b2=1, bab=a-1 >

C1
C2
60C2
60C2
C3
C5
C4
30C22
30C22
C6
20S3
20S3
C10
12D5
12D5
C15
C8
15D4
15D4
C12
10D6
10D6
C20
6D10
6D10
C30
4D15
4D15
15D8
C24
5D12
5D12
C40
3D20
3D20
C60
2D30
2D30
5D24
3D40
D60
D60
C120
D120

Smallest permutation representation of D120
►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)

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

D120 is a maximal subgroup of
 C3⋊D80  C5⋊D48  C24.D10  Dic12⋊D5  D240  C48⋊D5  C15⋊7D16  C8.6D30  D5×D24  C24⋊D10  S3×D40  C40⋊1D6  D120⋊C2  D120⋊5C2  C40.69D6  C8⋊D30  D8×D15  Q8⋊3D30  D120⋊8C2
D120 is a maximal quotient of
 D240  C48⋊D5  Dic120  C120⋊9C4  D60⋊8C4

63 conjugacy classes

class 1 2A2B2C 3  4 5A5B 6 8A8B10A10B12A12B15A15B15C15D20A20B20C20D24A24B24C24D30A30B30C30D40A···40H60A···60H120A···120P
order12223455688101012121515151520202020242424243030303040···4060···60120···120
size1160602222222222222222222222222222···22···22···2

63 irreducible representations

dim11122222222222222
type+++++++++++++++++
imageC1C2C2S3D4D5D6D8D10D12D15D20D24D30D40D60D120
kernelD120C120D60C40C30C24C20C15C12C10C8C6C5C4C3C2C1
# reps112112122244448816

Matrix representation of D120 ►in GL2(𝔽241) generated by

10499
14217
,
10499
234137
G:=sub<GL(2,GF(241))| [104,142,99,17],[104,234,99,137] >;
 

D120 in GAP, Magma, Sage, TeX 

D_{120}
 
% in TeX
 
G:=Group("D120");
 
// GroupNames label
 
G:=SmallGroup(240,68);
 
// by ID
 
G=gap.SmallGroup(240,68);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,73,79,218,50,964,6917]);
 
// Polycyclic
 
G:=Group<a,b|a^120=b^2=1,b*a*b=a^-1>;
 
// generators/relations
 

Export

Subgroup lattice of D120 in TeX

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