Finite Group Databasebeta

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Cayley diagram

C7×(C2×D3)⋊C4 #66615

order336
isoclinismisoclinic class of Dic6
automorphismAut(G) = C25×C3×D3 (order 576) · Inn(G) = C2×D3 · Out(G) = C24×C3
homologyAbelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C22

Aliases

direct productC7×(C2×D3)⋊C4
unambiguous semidirect product(C2×D3)⋊(C4×C7)

Splittings

FittingC2×C3×C4×C7 ⋊ C2
HallC3×C7 ⋊ C22⋊C4
SylowC3 ⋊ C7×C22⋊C4

Subgroups

subgroupG/H
centerC22×C7C2×D3
commutatorC2×C3C2×C4×C7
FrattiniC22C2×D3×C7
FittingC2×C3×C4×C7C2
socleC22×C3×C7C22
prime radicalC22×D3×C7C2
cosocleC22×C3C22×C7
hypercenterC2×C4×C7D3
hypocenterC3C7×C22⋊C4
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C7 ◁ C2×C4×C7

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 168

A minimal faithful permutation representation — 2 generators on 18 points:

(1,2,3)(4,7,6,5)(9,11)(12,13,14,15,16,17,18);(2,3)(8,9)(10,11)

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10620888, #10620903, #10620904, …
maximal normal subgroup of#10620888, #10620903, #10620904, …
maximal quotient ofC7×#805, C7×#806, C7×#809, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 84);