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Cayley diagram

C4×D3×C7 #4260

order168
isoclinismisoclinic class of D3
automorphismAut(G) = C23×C3×D3 (order 144) · Inn(G) = D3 · Out(G) = C23×C3
homologyAbelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C2

Aliases

direct productC4×D3×C7

Splittings

FittingC3×C4×C7 ⋊ C2
derived, SylowC3 ⋊ C2×C4×C7
HallC3×C7 ⋊ C2×C4

Subgroups

subgroupG/H
center, hypercenterC4×C7D3
commutator, hypocenterC3C2×C4×C7
FrattiniC2C2×D3×C7
FittingC3×C4×C7C2
socleC2×C3×C7C22
prime radicalC2×D3×C7C2
cosocleC2×C3C22×C7
LCSG ▷ C3
UCSC1 ◁ C4×C7

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 84

A minimal faithful permutation representation — 2 generators on 14 points:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×C7⋊C8, #66558, D3×C7×C8, …
maximal normal subgroup ofD3×C7⋊C8, #66558, D3×C7×C8, …
maximal quotient ofD3×C7×C8, C7×ΓL(1,𝔽25), C4×C7×Dic3, …
automorphism group ofC22×C29, D3×C29
outer automorphism group ofC22×C29, Q8×C29

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(168, 30);