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Cayley diagram

D7×S4 #66743

order336
isoclinismisoclinic class of D7×S4
automorphismAut(G) = S4×AGL(1,𝔽7) (order 1008) · Inn(G) = G · Out(G) = C3
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C22

Aliases

direct productD7×S4
unambiguous semidirect product(C7×S4)⋊C2
unambiguous semidirect product(C7⋊S4)⋊C2
unambiguous semidirect product(A4×D7)⋊C2

Splittings

Fitting, socleC22×C7 ⋊ C2×D3
derivedC7×A4 ⋊ C22
SylowC7 ⋊ C2×S4

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, cosocle, hypocenterC7×A4C22
Fitting, socleC22×C7C2×D3
LCSG ▷ C7×A4
UCSC1

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 11
transitive degreeμt(G) = 28

A minimal faithful permutation representation — 2 generators on 11 points:

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

Conjugacy classes & character table

25 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×D7×S4, S4×AGL(1,𝔽7), C3×D7×S4, …
maximal normal subgroup ofC2×D7×S4, S4×AGL(1,𝔽7), C3×D7×S4, …
maximal quotient of#10621469, #10621470, #10621471, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 212);