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Cayley diagram

C2×C3×AGL(1,𝔽9) #89104

order432
isoclinismisoclinic class of AGL(1,𝔽9)
automorphismAut(G) = C22×AΓL(1,𝔽9) (order 576) · Inn(G) = AGL(1,𝔽9) · Out(G) = C23
homologyAbelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C2

Aliases

direct productC2×C3×AGL(1,𝔽9)

Splittings

Fitting, socleC2×C33 ⋊ C8
derivedC32 ⋊ C2×C3×C8
SylowC33 ⋊ C2×C8

Subgroups

subgroupG/H
center, hypercenterC2×C3AGL(1,𝔽9)
commutator, hypocenterC32C2×C3×C8
FrattiniC1G
Fitting, socleC2×C33C8
prime radicalC2×C3×C3⋊D3C4
cosocle#157C22×C3
LCSG ▷ C32
UCSC1 ◁ C2×C3

58 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 2 generators on 14 points:

(3,4,5)(7,8,9,12,10,13,14,11);(1,2)(6,9,14)(7,12,8)(10,13,11)

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic3×AGL(1,𝔽9), (C2×D3)⋊AGL(1,𝔽9), Dic3⋊AGL(1,𝔽9), …
maximal normal subgroup ofDic3×AGL(1,𝔽9), (C2×D3)⋊AGL(1,𝔽9), Dic3⋊AGL(1,𝔽9), …
maximal quotient ofC3×#64213, C3×#64214, C3×C4×AGL(1,𝔽9), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 751);