Finite Group Databasebeta

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Cayley diagram

D9×SL(2,𝔽3) #88617

order432
isoclinismisoclinic class of #88615
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productD9×SL(2,𝔽3)

Splittings

Fitting, derivedQ8×C9 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC2A4×D9
commutator, Fitting, cosocle, hypocenterQ8×C9C2×C3
Frattini, socleC2×C3D3×A4
LCSG ▷ Q8×C9
UCSC1 ◁ C2

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 72

A minimal faithful permutation representation — 2 generators on 17 points:

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

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD9×#226, #11722233, #11722234, …
maximal normal subgroup ofD9×#226, #11722233, #11722234, …
maximal quotient ofSL(2,𝔽3)×Dic9, C2×D9×SL(2,𝔽3), SL(2,𝔽3)×D27, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 264);