Finite Group Databasebeta

← Prev (#88593) Next (#88595) →

Cayley diagram

C9×GL(2,𝔽3) #88594

order432
isoclinismisoclinic class of #226
automorphismAut(G) = C22×C3×S4 (order 288) · Inn(G) = S4 · Out(G) = C22×C3
homologyAbelianization (H1) = C2×C9 · Schur multiplier (H2) = C1

Aliases

direct productC9×GL(2,𝔽3)

Splittings

FittingQ8×C9 ⋊ D3
derivedSL(2,𝔽3) ⋊ C2×C9

Subgroups

subgroupG/H
center, hypercenterC2×C9S4
commutator, hypocenterSL(2,𝔽3)C2×C9
Frattini, socleC2×C3C3×S4
FittingQ8×C9D3
prime radicalC3×GL(2,𝔽3)C3
cosocleC3×SL(2,𝔽3)C2×C3
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C9

54 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:

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

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11722231, #11722233, #11722235, …
maximal normal subgroup of#11722231, #11722233, #11722235, …
maximal quotient ofC9×#859, C2×C9×GL(2,𝔽3), C27×GL(2,𝔽3), …
outer automorphism group of#6903

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 241);