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Cayley diagram

C22×C3×A4 #3850

order144
isoclinismisoclinic class of A4
automorphismAut(G) = D32×S4 (order 864) · Inn(G) = A4 · Out(G) = C2×D32
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C22×C3

Aliases

direct productC22×C3×A4

Splittings

Fitting, socleC24×C3 ⋊ C3
derivedC22 ⋊ C22×C32
SylowC24 ⋊ C32

Subgroups

subgroupG/H
center, hypercenterC22×C3A4
commutator, cosocle, hypocenterC22C22×C32
FrattiniC1G
Fitting, socleC24×C3C3
LCSG ▷ C22
UCSC1 ◁ C22×C3

98 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:

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

derived subgroup of(C3⋊S4)⋊A4, A4×C3⋊S4, #11920641, …
maximal subgroup ofC2×C3×SL(2,ℤ/4), C3×GL(2,ℤ/4), C2×C3⋊SL(2,ℤ/4), …
maximal normal subgroup ofC2×C3×SL(2,ℤ/4), C3×GL(2,ℤ/4), C2×C3⋊SL(2,ℤ/4), …
maximal quotient ofC2×C3×C4×A4, C3×D4×A4, C22×C3×SL(2,𝔽3), …
outer automorphism group of#6439, C7×#126, C9×#126

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 193);