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Cayley diagram

C2×C3×GO+(4,𝔽2) #89107

order432
isoclinismisoclinic class of GO+(4,𝔽2)
automorphismAut(G) = C2×#10596417 (order 1152) · Inn(G) = GO+(4,𝔽2) · Out(G) = C2×D4
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C23

Aliases

direct productC2×C3×GO+(4,𝔽2)
unambiguous semidirect product#157⋊(C22×C3)
unambiguous semidirect productD32⋊(C22×C3)

Splittings

Fitting, socleC2×C33 ⋊ D4
SylowC33 ⋊ C2×D4

Subgroups

subgroupG/H
center, hypercenterC2×C3GO+(4,𝔽2)
commutator, cosocleC3⋊D3C23×C3
FrattiniC1G
Fitting, socleC2×C33D4
hypocenterC32C2×C3×D4
LCSG ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C2×C3

192 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 11
transitive degreeμt(G) = 24

A minimal faithful permutation representation — 3 generators on 11 points:

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

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×GO+(4,𝔽2)⋊C4, GO+(4,𝔽2)⋊Dic3, Dic3×GO+(4,𝔽2), …
maximal normal subgroup ofC3×GO+(4,𝔽2)⋊C4, GO+(4,𝔽2)⋊Dic3, Dic3×GO+(4,𝔽2), …
maximal quotient ofC3×D32⋊Q8, C3×#64221, C3×#64222, …
automorphism group ofD32×C7, D32×C9
outer automorphism group ofC7×H(5,𝔽2)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 754);