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Cayley diagram

C4×C7×A4 #66699

order336
isoclinismisoclinic class of A4
automorphismAut(G) = C22×C3×S4 (order 288) · Inn(G) = A4 · Out(G) = C23×C3
homologyAbelianization (H1) = C3×C4×C7 · Schur multiplier (H2) = C2

Aliases

direct productC4×C7×A4
unambiguous semidirect product(C22×C4)⋊(C3×C7)

Splittings

Fitting, HallC22×C4×C7 ⋊ C3
derivedC22 ⋊ C3×C4×C7
SylowC22×C4 ⋊ C3×C7

Subgroups

subgroupG/H
center, hypercenterC4×C7A4
commutator, hypocenterC22C3×C4×C7
FrattiniC2C2×C7×A4
FittingC22×C4×C7C3
socleC23×C7C2×C3
prime radicalC2×C7×A4C2
cosocleC23C2×C3×C7
LCSG ▷ C22
UCSC1 ◁ C4×C7

38 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 84

A minimal faithful permutation representation — 2 generators on 15 points:

(1,2,3,4)(6,7,8)(9,10,11,12,13,14,15);(5,8)(6,7)

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

derived subgroup of#13057036, #13057037, #13057038, …
maximal subgroup ofC7×A4⋊C8, C7⋊(A4⋊C8), A4×C7⋊C8, …
maximal normal subgroup ofC7×A4⋊C8, C7⋊(A4⋊C8), A4×C7⋊C8, …
maximal quotient ofC4×C7×SL(2,𝔽3), C7×C8×A4, C7×#131⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 168);