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G = D48order 96 = 25·3

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D48, C31D16, C481C2, C161S3, C6.1D8, D241C2, C8.13D6, C4.1D12, C2.3D24, C12.24D4, C24.14C22, sometimes denoted D96 or Dih48 or Dih96, SmallGroup(96,6)

Series: Derived Chief Lower central Upper central

C1C24 — D48
C1C3C6C12C24D24 — D48
C3C6C12C24 — D48
C1C2C4C8C16

Generators and relations for D48
 G = < a,b | a48=b2=1, bab=a-1 >

24C2
24C2
12C22
12C22
8S3
8S3
6D4
6D4
4D6
4D6
3D8
3D8
2D12
2D12
3D16

Character table of D48

 class 12A2B2C3468A8B12A12B16A16B16C16D24A24B24C24D48A48B48C48D48E48F48G48H
 size 11242422222222222222222222222
ρ1111111111111111111111111111    trivial
ρ2111-11111111-1-1-1-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ311-1-111111111111111111111111    linear of order 2
ρ411-111111111-1-1-1-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ52200-12-122-1-1-2-2-2-2-1-1-1-111111111    orthogonal lifted from D6
ρ62200222-2-2220000-2-2-2-200000000    orthogonal lifted from D4
ρ72200-12-122-1-12222-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ822002-2200-2-22-2-220000-222-222-2-2    orthogonal lifted from D8
ρ922002-2200-2-2-222-200002-2-22-2-222    orthogonal lifted from D8
ρ102200-12-1-2-2-1-1000011113-33-3-33-33    orthogonal lifted from D12
ρ112200-12-1-2-2-1-100001111-33-333-33-3    orthogonal lifted from D12
ρ122-20020-22-200165163ζ16151691615169ζ165163-2-222ζ1615169ζ165163ζ165163ζ161516916516316516316151691615169    orthogonal lifted from D16
ρ132-20020-2-2200ζ1615169ζ165163165163161516922-2-2ζ16516316151691615169ζ165163ζ1615169ζ1615169165163165163    orthogonal lifted from D16
ρ142-20020-22-200ζ1651631615169ζ1615169165163-2-22216151691651631651631615169ζ165163ζ165163ζ1615169ζ1615169    orthogonal lifted from D16
ρ152-20020-2-22001615169165163ζ165163ζ161516922-2-2165163ζ1615169ζ161516916516316151691615169ζ165163ζ165163    orthogonal lifted from D16
ρ162200-1-2-10011-222-23-3-33ζ87ζ385ζ385ζ83ζ328ζ328ζ87ζ328785ζ32ζ83ζ3838ζ3ζ83ζ328ζ328ζ87ζ328785ζ32ζ83ζ3838ζ3ζ87ζ385ζ385    orthogonal lifted from D24
ρ172200-1-2-10011-222-2-333-3ζ83ζ3838ζ3ζ87ζ328785ζ32ζ83ζ328ζ328ζ87ζ385ζ385ζ87ζ328785ζ32ζ83ζ328ζ328ζ87ζ385ζ385ζ83ζ3838ζ3    orthogonal lifted from D24
ρ182200-1-2-100112-2-223-3-33ζ87ζ328785ζ32ζ83ζ3838ζ3ζ87ζ385ζ385ζ83ζ328ζ328ζ83ζ3838ζ3ζ87ζ385ζ385ζ83ζ328ζ328ζ87ζ328785ζ32    orthogonal lifted from D24
ρ192200-1-2-100112-2-22-333-3ζ83ζ328ζ328ζ87ζ385ζ385ζ83ζ3838ζ3ζ87ζ328785ζ32ζ87ζ385ζ385ζ83ζ3838ζ3ζ87ζ328785ζ32ζ83ζ328ζ328    orthogonal lifted from D24
ρ202-200-1012-2-33165163ζ1615169ζ16716ζ165163ζ1614ζ316141610ζ3ζ166ζ3162ζ3162ζ166ζ32166162ζ32ζ1614ζ321610ζ321610ζ167ζ316716ζ3ζ165ζ32163ζ32163ζ1613ζ3216131611ζ32ζ167ζ3216716ζ32ζ165ζ3165163ζ3ζ1613ζ31611ζ31611ζ167ζ316ζ316ζ167ζ3216ζ3216    orthogonal faithful
ρ212-200-101-223-3ζ1615169ζ165163165163ζ16716ζ166ζ32166162ζ32ζ1614ζ321610ζ321610ζ1614ζ316141610ζ3ζ166ζ3162ζ3162ζ165ζ32163ζ32163ζ167ζ3216ζ3216ζ167ζ316ζ316ζ1613ζ3216131611ζ32ζ167ζ316716ζ3ζ167ζ3216716ζ32ζ1613ζ31611ζ31611ζ165ζ3165163ζ3    orthogonal faithful
ρ222-200-101-22-33ζ1615169ζ165163165163ζ16716ζ1614ζ321610ζ321610ζ166ζ32166162ζ32ζ166ζ3162ζ3162ζ1614ζ316141610ζ3ζ1613ζ3216131611ζ32ζ167ζ316ζ316ζ167ζ3216ζ3216ζ165ζ32163ζ32163ζ167ζ3216716ζ32ζ167ζ316716ζ3ζ165ζ3165163ζ3ζ1613ζ31611ζ31611    orthogonal faithful
ρ232-200-101-223-3ζ16716165163ζ165163ζ1615169ζ166ζ32166162ζ32ζ1614ζ321610ζ321610ζ1614ζ316141610ζ3ζ166ζ3162ζ3162ζ165ζ3165163ζ3ζ167ζ316716ζ3ζ167ζ3216716ζ32ζ1613ζ31611ζ31611ζ167ζ3216ζ3216ζ167ζ316ζ316ζ1613ζ3216131611ζ32ζ165ζ32163ζ32163    orthogonal faithful
ρ242-200-101-22-33ζ16716165163ζ165163ζ1615169ζ1614ζ321610ζ321610ζ166ζ32166162ζ32ζ166ζ3162ζ3162ζ1614ζ316141610ζ3ζ1613ζ31611ζ31611ζ167ζ3216716ζ32ζ167ζ316716ζ3ζ165ζ3165163ζ3ζ167ζ316ζ316ζ167ζ3216ζ3216ζ165ζ32163ζ32163ζ1613ζ3216131611ζ32    orthogonal faithful
ρ252-200-1012-2-33ζ165163ζ16716ζ1615169165163ζ1614ζ316141610ζ3ζ166ζ3162ζ3162ζ166ζ32166162ζ32ζ1614ζ321610ζ321610ζ167ζ3216ζ3216ζ165ζ3165163ζ3ζ1613ζ31611ζ31611ζ167ζ316ζ316ζ165ζ32163ζ32163ζ1613ζ3216131611ζ32ζ167ζ3216716ζ32ζ167ζ316716ζ3    orthogonal faithful
ρ262-200-1012-23-3ζ165163ζ16716ζ1615169165163ζ166ζ3162ζ3162ζ1614ζ316141610ζ3ζ1614ζ321610ζ321610ζ166ζ32166162ζ32ζ167ζ316ζ316ζ1613ζ31611ζ31611ζ165ζ3165163ζ3ζ167ζ3216ζ3216ζ1613ζ3216131611ζ32ζ165ζ32163ζ32163ζ167ζ316716ζ3ζ167ζ3216716ζ32    orthogonal faithful
ρ272-200-1012-23-3165163ζ1615169ζ16716ζ165163ζ166ζ3162ζ3162ζ1614ζ316141610ζ3ζ1614ζ321610ζ321610ζ166ζ32166162ζ32ζ167ζ3216716ζ32ζ1613ζ3216131611ζ32ζ165ζ32163ζ32163ζ167ζ316716ζ3ζ1613ζ31611ζ31611ζ165ζ3165163ζ3ζ167ζ3216ζ3216ζ167ζ316ζ316    orthogonal faithful

Smallest permutation representation of D48
On 48 points
Generators in S48
(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)
(1 48)(2 47)(3 46)(4 45)(5 44)(6 43)(7 42)(8 41)(9 40)(10 39)(11 38)(12 37)(13 36)(14 35)(15 34)(16 33)(17 32)(18 31)(19 30)(20 29)(21 28)(22 27)(23 26)(24 25)

G:=sub<Sym(48)| (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), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)>;

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), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25) );

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)], [(1,48),(2,47),(3,46),(4,45),(5,44),(6,43),(7,42),(8,41),(9,40),(10,39),(11,38),(12,37),(13,36),(14,35),(15,34),(16,33),(17,32),(18,31),(19,30),(20,29),(21,28),(22,27),(23,26),(24,25)])

D48 is a maximal subgroup of
D96  C32⋊S3  C3⋊D32  C3⋊SD64  D487C2  C16⋊D6  S3×D16  D48⋊C2  D485C2  D144  C3⋊D48  C325D16  C5⋊D48  D240
D48 is a maximal quotient of
D96  C32⋊S3  Dic48  C485C4  C2.D48  D144  C3⋊D48  C325D16  C5⋊D48  D240

Matrix representation of D48 in GL2(𝔽47) generated by

3716
312
,
238
1645
G:=sub<GL(2,GF(47))| [37,31,16,2],[2,16,38,45] >;

D48 in GAP, Magma, Sage, TeX

D_{48}
% in TeX

G:=Group("D48");
// GroupNames label

G:=SmallGroup(96,6);
// by ID

G=gap.SmallGroup(96,6);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,73,79,218,122,579,69,2309]);
// Polycyclic

G:=Group<a,b|a^48=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D48 in TeX
Character table of D48 in TeX

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