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Cayley diagram

D96 #4448

order192
isoclinismisoclinic class of D96
automorphismAut(G) = D3×AGL(1,ℤ/32) (order 3072) · Inn(G) = D48 · Out(G) = C22×C8
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD96

Splittings

FittingC3×C32 ⋊ C2
SylowC3 ⋊ D32

Subgroups

subgroupG/H
centerC2D48
commutator, cosocleC3×C16C22
FrattiniC16C2×D3
FittingC3×C32C2
socleC2×C3D16
hypercenterC32D3
hypocenterC3D32
LCSG ▷ C3×C16 ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32

34 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 35
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 2 generators on 35 points:

(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);(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(34,35)

Polycyclic presentation:

2^6,3|b^2=cd,c^2=de,d^2=ef,e^2=f,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^2

Conjugacy classes & character table

51 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD192, #67434, #67607, …
maximal normal subgroup ofD192, #67434, #67607, …
maximal quotient ofD192, #67434, Dic96, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 7);