Finite Group Databasebeta

← Prev (#6298) Next (#6300) →

Cayley diagram

C32×C4×D3 #6299

order216
isoclinismisoclinic class of D3
automorphismAut(G) = C22×D3×GL(2,𝔽3) (order 1152) · Inn(G) = D3 · Out(G) = C22×GL(2,𝔽3)
homologyAbelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C2×C3

Aliases

direct productC32×C4×D3

Splittings

FittingC33×C4 ⋊ C2
derivedC3 ⋊ C2×C32×C4
SylowC33 ⋊ C2×C4

Subgroups

subgroupG/H
center, hypercenterC32×C4D3
commutator, hypocenterC3C2×C32×C4
FrattiniC2C2×C32×D3
FittingC33×C4C2
socleC2×C33C22
prime radicalC2×C32×D3C2
cosocleC2×C3C22×C32
LCSG ▷ C3
UCSC1 ◁ C32×C4

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 72

A minimal faithful permutation representation — 2 generators on 13 points:

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

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×D3×C3⋊C8, C3×#3711, D3×C3⋊(C3⋊C8), …
maximal normal subgroup ofC3×D3×C3⋊C8, C3×#3711, D3×C3⋊(C3⋊C8), …
maximal quotient ofC32×D3×C8, C32×ΓL(1,𝔽25), C32×C4×Dic3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 136);