Finite Group Databasebeta

← Prev (#66598) Next (#66600) →

Cayley diagram

C3×(C2×D7)⋊C4 #66599

order336
isoclinismisoclinic class of Dic14
automorphismAut(G) = C25×AGL(1,𝔽7) (order 1344) · Inn(G) = C2×D7 · Out(G) = C24×C3
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22

Aliases

direct productC3×(C2×D7)⋊C4

Splittings

FittingC2×C3×C4×C7 ⋊ C2
HallC3×C7 ⋊ C22⋊C4
SylowC7 ⋊ C3×C22⋊C4

Subgroups

subgroupG/H
centerC22×C3C2×D7
commutatorC2×C7C2×C3×C4
FrattiniC22C2×C3×D7
FittingC2×C3×C4×C7C2
socleC22×C3×C7C22
prime radicalC22×C3×D7C2
cosocleC22×C7C22×C3
hypercenterC2×C3×C4D7
hypocenterC7C3×C22⋊C4
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

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:

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

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10620915, #10620916, #10620918, …
maximal normal subgroup of#10620915, #10620916, #10620918, …
maximal quotient ofC3×#6379, C3×#6380, C3×#6383, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 68);