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Cayley diagram

C9×#41⋊C3 #88682

order432
isoclinismisoclinic class of SL(2,𝔽3)
automorphismAut(G) = C2×C3×D3×S4 (order 864) · Inn(G) = A4 · Out(G) = C22×C3×D3
homologyAbelianization (H1) = C2×C3×C9 · Schur multiplier (H2) = C3

Aliases

direct productC9×#41⋊C3

Splittings

FittingC9×#41 ⋊ C3
Sylow#41 ⋊ C3×C9

Subgroups

subgroupG/H
center, hypercenterC4×C9A4
commutator, hypocenterQ8C2×C3×C9
Frattini, socleC2×C3C2×C3×A4
FittingC9×#41C3
prime radicalC3×#41⋊C3C3
cosocleC3×Q8C2×C32
LCSG ▷ Q8
UCSC1 ◁ C4×C9

65 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 25
transitive degreeμt(G) = 144

A minimal faithful permutation representation — 2 generators on 25 points:

(1,2,5,8)(3,7,11,14,4,13,10,16,9,6,12,15);(1,9,5,11)(2,13,8,15)(3,12,10,4)(6,16,14,7)(17,18,20,19,21,23,22,24,25)

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

derived subgroup of#13146239, #13146240, #13146241, …
maximal subgroup ofC9×GU(2,𝔽3), C9⋊GU(2,𝔽3), #11720753, …
maximal normal subgroup ofC9×GU(2,𝔽3), C9⋊GU(2,𝔽3), #11720753, …
maximal quotient ofC4×C9×SL(2,𝔽3), C2×C9×#41⋊C3, C27×#41⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 329);