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Cayley diagram

A4×D7 #4278

order168
isoclinismisoclinic class of A4×D7
automorphismAut(G) = S4×AGL(1,𝔽7) (order 1008) · Inn(G) = G · Out(G) = C2×C3
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C2

Aliases

direct productA4×D7

Splittings

Fitting, socle, derivedC22×C7 ⋊ C2×C3
HallC22×D7 ⋊ C3
SylowC7 ⋊ C2×A4

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC22×C7C2×C3
LCSG ▷ C22×C7
UCSC1

24 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:

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

Conjugacy classes & character table

20 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD7×S4, C2×A4×D7, A4×AGL(1,𝔽7), …
maximal normal subgroup ofD7×S4, C2×A4×D7, A4×AGL(1,𝔽7), …
maximal quotient of#66662, D7×SL(2,𝔽3), A4×Dic7, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(168, 48);