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Cayley diagram

D3×AGL(1,𝔽9) #89089

order432
isoclinismisoclinic class of D3×AGL(1,𝔽9)
automorphismAut(G) = D3×AΓL(1,𝔽9) (order 864) · Inn(G) = G · Out(G) = C2
homologyAbelianization (H1) = C2×C8 · Schur multiplier (H2) = C2

Aliases

direct productD3×AGL(1,𝔽9)

Splittings

Fitting, socle, derived, SylowC33 ⋊ C2×C8

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC33C2×C8
prime radicalD3×C3⋊D3C4
cosocleC3×#157C22
LCSG ▷ C33
UCSC1

58 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 24

A minimal faithful permutation representation — 2 generators on 12 points:

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

Conjugacy classes & character table

27 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×AΓL(1,𝔽9), C2×D3×AGL(1,𝔽9), D9×AGL(1,𝔽9), …
maximal normal subgroup ofD3×AΓL(1,𝔽9), C2×D3×AGL(1,𝔽9), C3×D3×AGL(1,𝔽9)
maximal quotient ofDic3×AGL(1,𝔽9), (C2×D3)⋊AGL(1,𝔽9), Dic3⋊AGL(1,𝔽9), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 736);