Finite Group Databasebeta

← Prev (#66706) Next (#66708) →

Cayley diagram

C2×C3×D28 #66707

order336
isoclinismisoclinic class of Dic14
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C23

Aliases

direct productC2×C3×D28

Splittings

FittingC2×C3×C4×C7 ⋊ C2
HallC3×C7 ⋊ C2×D4
SylowC7 ⋊ C2×C3×D4

Subgroups

subgroupG/H
centerC22×C3C2×D7
commutator, cosocleC2×C7C23×C3
FrattiniC2C22×C3×D7
FittingC2×C3×C4×C7C2
socleC22×C3×C7C22
hypercenterC2×C3×C4D7
hypocenterC7C2×C3×D4
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C3 ◁ C2×C3×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)(7,9)(11,12)(13,14)(15,16);(1,2)

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10620493, #10620499, #10620500, …
maximal normal subgroup of#10620493, #10620499, #10620500, …
maximal quotient ofC3×#6432, C3×C4×D28, C3×#6437, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 176);