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Cayley diagram

C8×A4 #866

order96
isoclinismisoclinic class of A4
automorphismAut(G) = C22×S4 (order 96) · Inn(G) = A4 · Out(G) = C23
homologyAbelianization (H1) = C3×C8 · Schur multiplier (H2) = C2

Aliases

direct productC8×A4
unambiguous semidirect product(C22×C8)⋊C3

Splittings

Fitting, SylowC22×C8 ⋊ C3
derivedC22 ⋊ C3×C8

Subgroups

subgroupG/H
center, hypercenterC8A4
commutator, hypocenterC22C3×C8
FrattiniC4C2×A4
FittingC22×C8C3
socleC23C3×C4
prime radicalC2×A4C4
cosocleC22×C4C2×C3
LCSG ▷ C22
UCSC1 ◁ C8

26 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 24

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

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

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

derived subgroup of#73035, #73036, #73037, …
maximal subgroup ofA4⋊C16, A4×C16, #5398, …
maximal normal subgroup ofA4⋊C16, A4×C16, #5398, …
maximal quotient ofC8×SL(2,𝔽3), A4×C16, #504⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 73);