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Cayley diagram

C4×A4 #229

order48
isoclinismisoclinic class of A4
automorphismAut(G) = C2×S4 (order 48) · Inn(G) = A4 · Out(G) = C22
holomorphG ⋊ Aut(G) = D4×A4⋊S4
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C2

Aliases

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

Splittings

Fitting, SylowC22×C4 ⋊ C3
derivedC22 ⋊ C3×C4

Subgroups

subgroupG/H
center, hypercenterC4A4
commutator, hypocenterC22C3×C4
FrattiniC2C2×A4
FittingC22×C4C3
socle, cosocleC23C2×C3
prime radicalC2×A4C2
LCSG ▷ C22
UCSC1 ◁ C4

19 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 8
transitive degreeμt(G) = 12

A minimal faithful permutation representation — 2 generators on 8 points:

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

Conjugacy classes & character table

16 conjugacy classes — view the full character table.

Relationships

derived subgroup of#5398, #5401, #5402, …
maximal subgroup ofA4⋊C8, C8×A4, A4⋊Q8, …
maximal normal subgroup ofA4⋊C8, C8×A4, A4⋊Q8, …
maximal quotient ofC4×SL(2,𝔽3), C8×A4, #131⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(48, 31);