Finite Group Databasebeta

← Prev (#89049) Next (#89051) →

Cayley diagram

C32×C4×A4 #89050

order432
isoclinismisoclinic class of A4
automorphism?
homologyAbelianization (H1) = C33×C4 · Schur multiplier (H2) = C2×C33

Aliases

direct productC32×C4×A4
unambiguous semidirect product(C22×C4)⋊C33
unambiguous semidirect product(C22×C3×C4)⋊C32

Splittings

FittingC22×C32×C4 ⋊ C3
derivedC22 ⋊ C33×C4
SylowC22×C4 ⋊ C33

Subgroups

subgroupG/H
center, hypercenterC32×C4A4
commutator, hypocenterC22C33×C4
FrattiniC2C2×C32×A4
FittingC22×C32×C4C3
socleC23×C32C2×C3
prime radicalC2×C32×A4C2
cosocleC23C2×C33
LCSG ▷ C22
UCSC1 ◁ C32×C4

210 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:

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

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

derived subgroup of#13165927, #13165928, #13165929, …
maximal subgroup ofC32×A4⋊C8, C3×C3⋊(A4⋊C8), C3⋊(C3⋊(A4⋊C8)), …
maximal normal subgroup ofC32×A4⋊C8, C3×C3⋊(A4⋊C8), C3⋊(C3⋊(A4⋊C8)), …
maximal quotient ofC32×C4×SL(2,𝔽3), C32×C8×A4, C32×#131⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 697);