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Cayley diagram

C2×C4×A4 #989

order96
isoclinismisoclinic class of A4
automorphismAut(G) = D4×S4 (order 192) · Inn(G) = A4 · Out(G) = C2×D4
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22

Aliases

direct productC2×C4×A4
unambiguous semidirect product(C23×C4)⋊C3

Splittings

Fitting, SylowC23×C4 ⋊ C3
derivedC22 ⋊ C2×C3×C4

Subgroups

subgroupG/H
center, hypercenterC2×C4A4
commutator, hypocenterC22C2×C3×C4
FrattiniC2C22×A4
FittingC23×C4C3
socleC24C2×C3
prime radicalC22×A4C2
cosocleC23C22×C3
LCSG ▷ C22
UCSC1 ◁ C2×C4

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 10
transitive degreeμt(G) = 24

A minimal faithful permutation representation — 2 generators on 10 points:

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

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

derived subgroup of#11711019, #11711020, #11711021, …
maximal subgroup ofC2×A4⋊C8, #5409, C4×SL(2,ℤ/4), …
maximal normal subgroup ofC2×A4⋊C8, #5409, C4×SL(2,ℤ/4), …
maximal quotient ofC42×A4, A4×C22⋊C4, A4×C4⋊C4, …
outer automorphism group of#4022, C5×#126

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 196);