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Cayley diagram

C2×C7×A4 #4282

order168
isoclinismisoclinic class of A4
automorphismAut(G) = C2×C3×S4 (order 144) · Inn(G) = A4 · Out(G) = C22×C3
holomorphG ⋊ Aut(G) = C2×AGL(1,𝔽7)×A4⋊S4
homologyAbelianization (H1) = C2×C3×C7 · Schur multiplier (H2) = C2

Aliases

direct productC2×C7×A4

Splittings

Fitting, socle, HallC23×C7 ⋊ C3
derivedC22 ⋊ C2×C3×C7
SylowC23 ⋊ C3×C7

Subgroups

subgroupG/H
center, hypercenterC2×C7A4
commutator, cosocle, hypocenterC22C2×C3×C7
FrattiniC1G
Fitting, socleC23×C7C3
LCSG ▷ C22
UCSC1 ◁ C2×C7

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 42

A minimal faithful permutation representation — 2 generators on 13 points:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

derived subgroup of#10621477, #10621480, #10621482, …
maximal subgroup ofC7×SL(2,ℤ/4), C7⋊SL(2,ℤ/4), A4×Dic7, …
maximal normal subgroup ofC7×SL(2,ℤ/4), C7⋊SL(2,ℤ/4), A4×Dic7, …
maximal quotient ofC4×C7×A4, C2×C7×SL(2,𝔽3), C7×#41⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(168, 52);