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Cayley diagram

C32×C22⋊A4 #89124

order432
isoclinismisoclinic class of C22⋊A4
automorphism?
homologyAbelianization (H1) = C33 · Schur multiplier (H2) = C24×C33

Aliases

direct productC32×C22⋊A4

Splittings

Fitting, socleC24×C32 ⋊ C3
derived, SylowC24 ⋊ C33

Subgroups

subgroupG/H
center, hypercenterC32C22⋊A4
commutator, cosocle, hypocenterC24C33
FrattiniC1G
Fitting, socleC24×C32C3
LCSG ▷ C24
UCSC1 ◁ C32

316 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 108

A minimal faithful permutation representation — 3 generators on 14 points:

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

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

derived subgroup ofC3⋊(C3⋊#1020), C3⋊(C3⋊#4625), #13165839, …
maximal subgroup ofC3×C3⋊(#120⋊C3), C3⋊(C3⋊(#120⋊C3)), C32×#120⋊C3, …
maximal normal subgroup ofC3×C3⋊(#120⋊C3), C3⋊(C3⋊(#120⋊C3)), C32×#120⋊C3, …
maximal quotient ofC32×C22⋊SL(2,𝔽3), C32×#997, C2×C32×C22⋊A4, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 771);