Finite Group Databasebeta

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Cayley diagram

C2×C7×D12 #66717

order336
isoclinismisoclinic class of Dic6
automorphism?
homologyAbelianization (H1) = C23×C7 · Schur multiplier (H2) = C23

Aliases

direct productC2×C7×D12

Splittings

FittingC2×C3×C4×C7 ⋊ C2
HallC3×C7 ⋊ C2×D4
SylowC3 ⋊ C2×C7×D4

Subgroups

subgroupG/H
centerC22×C7C2×D3
commutator, cosocleC2×C3C23×C7
FrattiniC2C22×D3×C7
FittingC2×C3×C4×C7C2
socleC22×C3×C7C22
hypercenterC2×C4×C7D3
hypocenterC3C2×C7×D4
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C7 ◁ C2×C4×C7

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 168

A minimal faithful permutation representation — 3 generators on 16 points:

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

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10620494, #10620501, #10620502, …
maximal normal subgroup of#10620494, #10620501, #10620502, …
maximal quotient ofC7×#869, C4×C7×D12, C7×#874, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 186);