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Cayley diagram

C22×C4×A4 #5937

order192
isoclinismisoclinic class of A4
automorphismAut(G) = S4×#5934 (order 4608) · Inn(G) = A4 · Out(G) = C2×#5934
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24

Aliases

direct productC22×C4×A4

Splittings

Fitting, SylowC24×C4 ⋊ C3
derivedC22 ⋊ C22×C3×C4

Subgroups

subgroupG/H
center, hypercenterC22×C4A4
commutator, hypocenterC22C22×C3×C4
FrattiniC2C23×A4
FittingC24×C4C3
socleC25C2×C3
prime radicalC23×A4C2
cosocleC23C23×C3
LCSG ▷ C22
UCSC1 ◁ C22×C4

317 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 3 generators on 12 points:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

derived subgroup of#72228216, #72268141, #72268159, …
maximal subgroup of#73141, A4⋊#95, A4×#95, …
maximal normal subgroup of#73141, A4⋊#95, A4×#95, …
maximal quotient ofC2×C42×A4, C2×A4×C22⋊C4, C2×A4×C4⋊C4, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1496);