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Cayley diagram

C22×C7×A4 #66752

order336
isoclinismisoclinic class of A4
automorphismAut(G) = C2×C3×D3×S4 (order 864) · Inn(G) = A4 · Out(G) = C22×C3×D3
homologyAbelianization (H1) = C22×C3×C7 · Schur multiplier (H2) = C22

Aliases

direct productC22×C7×A4

Splittings

Fitting, socle, HallC24×C7 ⋊ C3
derivedC22 ⋊ C22×C3×C7
SylowC24 ⋊ C3×C7

Subgroups

subgroupG/H
center, hypercenterC22×C7A4
commutator, cosocle, hypocenterC22C22×C3×C7
FrattiniC1G
Fitting, socleC24×C7C3
LCSG ▷ C22
UCSC1 ◁ C22×C7

78 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:

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

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

derived subgroup ofS4×C22⋊ΓL(1,𝔽8), A4×ΓL(1,𝔽8)×S4, A4×C7⋊S4, …
maximal subgroup ofC2×C7×SL(2,ℤ/4), C7×GL(2,ℤ/4), C2×C7⋊SL(2,ℤ/4), …
maximal normal subgroup ofC2×C7×SL(2,ℤ/4), C7×GL(2,ℤ/4), C2×C7⋊SL(2,ℤ/4), …
maximal quotient ofC2×C4×C7×A4, C7×D4×A4, C22×C7×SL(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 221);