Finite Group Databasebeta

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Cayley diagram

(C3⋊Dic3)⋊A4 #88625

order432
isoclinismisoclinic class of (C3⋊D3)⋊A4
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C2×C3

Aliases

unambiguous semidirect product(C3⋊Dic3)⋊A4

Splittings

derivedC22×C32 ⋊ C3×C4

Subgroups

subgroupG/H
center, hypercenterC2(C3⋊D3)⋊A4
commutator, hypocenterC22×C32C3×C4
FrattiniC2×C3D3×A4
Fitting, cosocleC23×C32C2×C3
socleC23×C3C3×D3
prime radicalC2×C22⋊H(3,𝔽3)C2
LCSG ▷ C22×C32
UCSC1 ◁ C2

92 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 36

A minimal faithful permutation representation — 2 generators on 17 points:

(1,2,3,4)(6,8,7)(10,15,17,12,13,16)(11,14);(5,8)(6,7)(9,10,12)(11,13,16)(14,17,15)

Polycyclic presentation:

2,3,2^3,3^2|a^2=c,f^a=f^2,g^a=g^2,d^b=de,e^b=d,f^b=fg

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#6258⋊C4, ((C3⋊S4)⋊C3)⋊C4, #11722273, …
maximal normal subgroup of#6258⋊C4, ((C3⋊S4)⋊C3)⋊C4, #11722273, …
maximal quotient of(C3⋊(C3⋊C8))⋊A4, #11720765, #11720767, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 272);