Finite Group Databasebeta

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Cayley diagram

(C3⋊SL(2,ℤ/4))⋊C3 #88604

order432
isoclinismisoclinic class of (C3⋊S4)⋊C3
automorphismAut(G) = C2×((C3⋊S4)⋊C3)⋊C2 (order 864) · Inn(G) = (C3⋊S4)⋊C3 · Out(G) = C22
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C2×C3

Aliases

unambiguous semidirect product(C3⋊SL(2,ℤ/4))⋊C3

Splittings

derivedC3×A4 ⋊ C3×C4

Subgroups

subgroupG/H
center, hypercenterC2(C3⋊S4)⋊C3
commutator, hypocenterC3×A4C3×C4
FrattiniC2×C3C3×S4
FittingC23×C32D3
socleC23×C3C3×D3
prime radicalC2×C22⋊H(3,𝔽3)C2
cosocleC2×C3×A4C2×C3
LCSG ▷ C3×A4
UCSC1 ◁ C2

82 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)(5,6,8,7)(10,15,17,12,13,16)(11,14);(6,7,8)(9,10,12)(11,13,16)(14,17,15)

Polycyclic presentation:

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of((C3⋊D3)⋊A4)⋊C4, #6258⋊C4, #11722271, …
maximal normal subgroup of((C3⋊D3)⋊A4)⋊C4, #6258⋊C4, #11722271, …
maximal quotient of(C3⋊(A4⋊C8))⋊C3, (C3⋊#859)⋊C3, (C3⋊GU(2,𝔽3))⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 251);