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Cayley diagram

C3×GO+(4,𝔽2) #6320

order216
isoclinismisoclinic class of GO+(4,𝔽2)
automorphismAut(G) = C2×AΓL(1,𝔽9) (order 288) · Inn(G) = GO+(4,𝔽2) · Out(G) = C22
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

direct productC3×GO+(4,𝔽2)
unambiguous semidirect product#157⋊(C2×C3)
unambiguous semidirect productD32⋊(C2×C3)

Splittings

Fitting, socle, SylowC33 ⋊ D4

Subgroups

subgroupG/H
center, hypercenterC3GO+(4,𝔽2)
commutator, cosocleC3⋊D3C22×C3
FrattiniC1G
Fitting, socleC33D4
hypocenterC32C3×D4
LCSG ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C3

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 9
transitive degreeμt(G) = 12

A minimal faithful permutation representation — 2 generators on 9 points:

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

Conjugacy classes & character table

27 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×AΓL(1,𝔽9), #89091, D3×GO+(4,𝔽2), …
maximal normal subgroup ofC3×AΓL(1,𝔽9), #89091, D3×GO+(4,𝔽2), …
maximal quotient ofC3×D32⋊C4, C3×#3773, C3×#3774, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 157);