Finite Group Databasebeta

← Prev (#66365) Next (#66367) →

Cayley diagram

#66366

order324
automorphismAut(G) = C3×A4⋊#272 (order 1944) · Inn(G) = C22⋊H(3,𝔽3) · Out(G) = C3×D3
homologyAbelianization (H1) = C32 · Schur multiplier (H2) = C2×C3

Aliases

No aliases for this group.

Splittings

FittingC22×3−1+2 ⋊ C3
SylowC22 ⋊ AΣL(1,𝔽27)

Subgroups

subgroupG/H
centerC3C22⋊H(3,𝔽3)
commutator, cosocleC22×C32C32
FrattiniC32C3×A4
FittingC22×3−1+2C3
socleC22×C3H(3,𝔽3)
hypercenter3−1+2A4
hypocenterC22AΣL(1,𝔽27)
LCSG ▷ C22×C32 ▷ C22×C3 ▷ C22
UCSC1 ◁ C3 ◁ C32 ◁ 3−1+2

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 36

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

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

Polycyclic presentation:

3^4,2^2|b^3=d,b^a=bc,e^a=f,f^a=ef,c^b=cd

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

derived subgroup of#66366⋊C2, #66366⋊C4, C2×#66366⋊C2, …
maximal subgroup of#66366⋊C2, C2×#66366, A4×AΣL(1,𝔽27), …
maximal normal subgroup of#66366⋊C2, C2×#66366, A4×AΣL(1,𝔽27), …
maximal quotient of#10619583, C2×#66366, #11757399, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 58);