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Cayley diagram

A4×C4⋊C4 #5436

order192
automorphismAut(G) = S4×#120 (order 768) · Inn(G) = C22×A4 · Out(G) = C2×D4
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22

Aliases

direct productA4×C4⋊C4
unambiguous semidirect productC4⋊(C4×A4)
unambiguous semidirect product(C22×C4⋊C4)⋊C3

Splittings

Fitting, SylowC22×C4⋊C4 ⋊ C3

Subgroups

subgroupG/H
center, FrattiniC22C22×A4
commutatorC23C2×C3×C4
FittingC22×C4⋊C4C3
socle, cosocleC24C22×C3
prime radicalC22×A4C22
hypercenterC4⋊C4A4
hypocenterC22C3×C4⋊C4
LCSG ▷ C23 ▷ C22
UCSC1 ◁ C22 ◁ C4⋊C4

113 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 12 points:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#73050, #73051, #73052, …
maximal normal subgroup of#73050, #73051, #73052, …
maximal quotient ofA4×#95, C4⋊C4×SL(2,𝔽3), #2351⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 995);