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Cayley diagram

C2×C3⋊AGL(1,𝔽9) #89105

order432
isoclinismisoclinic class of C3⋊AGL(1,𝔽9)
automorphismAut(G) = C2×D3×AΓL(1,𝔽9) (order 1728) · Inn(G) = C3⋊AGL(1,𝔽9) · Out(G) = C23
homologyAbelianization (H1) = C2×C8 · Schur multiplier (H2) = C2

Aliases

direct productC2×C3⋊AGL(1,𝔽9)
unambiguous semidirect product(C2×C3)⋊AGL(1,𝔽9)

Splittings

Fitting, socleC2×C33 ⋊ C8
derived, SylowC33 ⋊ C2×C8

Subgroups

subgroupG/H
center, hypercenterC2C3⋊AGL(1,𝔽9)
commutator, hypocenterC33C2×C8
FrattiniC1G
Fitting, socleC2×C33C8
prime radicalC2×C3×C3⋊D3C4
cosocleC3×#157C22
LCSG ▷ C33
UCSC1 ◁ C2

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:

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

Conjugacy classes & character table

30 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 of#11724237, #11724238, C4×C3⋊AGL(1,𝔽9), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 752);