Finite Group Databasebeta

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Cayley diagram

C2×D84 #66727

order336
isoclinismisoclinic class of Dic42
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C23

Aliases

direct productC2×D84

Splittings

FittingC2×C3×C4×C7 ⋊ C2
HallC3×C7 ⋊ C2×D4
SylowC7 ⋊ C2×D12
SylowC3 ⋊ C2×D28

Subgroups

subgroupG/H
centerC22C2×D21
commutator, cosocleC2×C3×C7C23
FrattiniC2C22×D21
FittingC2×C3×C4×C7C2
socleC22×C3×C7C22
hypercenterC2×C4D21
hypocenterC3×C7C2×D4
LCSG ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4

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

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10620495, #10620503, #10620504, …
maximal normal subgroup of#10620495, #10620503, #10620504, …
maximal quotient of#10621352, C4×D84, #10621357, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 196);