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Cayley diagram

C3×SL(2,ℤ/4) #3780

order144
isoclinismisoclinic class of S4
automorphismAut(G) = C22×S4 (order 96) · Inn(G) = S4 · Out(G) = C22
holomorphG ⋊ Aut(G) = D3×#72215293
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC3×SL(2,ℤ/4)
unambiguous semidirect productA4⋊(C3×C4)

Splittings

derivedA4 ⋊ C3×C4

Subgroups

subgroupG/H
center, hypercenterC2×C3S4
commutator, hypocenterA4C3×C4
FrattiniC2C3×S4
Fitting, socleC23×C3D3
prime radicalC2×C3×A4C2
cosocleC2×A4C2×C3
LCSG ▷ A4
UCSC1 ◁ C2×C3

42 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 11
transitive degreeμt(G) = 36

A minimal faithful permutation representation — 2 generators on 11 points:

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

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#64204, (C3⋊S4)⋊C4, D3×SL(2,ℤ/4), …
maximal normal subgroup of#64204, (C3⋊S4)⋊C4, D3×SL(2,ℤ/4), …
maximal quotient ofC3×A4⋊C8, C3×#859, C3×GU(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 123);