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Cayley diagram

C9×ΓL(1,𝔽25) #88463

order432
isoclinismisoclinic class of Dic6
automorphismAut(G) = C24×C3×D3 (order 288) · Inn(G) = C2×D3 · Out(G) = C23×C3
homologyAbelianization (H1) = C2×C4×C9 · Schur multiplier (H2) = C1

Aliases

direct productC9×ΓL(1,𝔽25)

Splittings

FittingC3×C8×C9 ⋊ C2
SylowC3×C9 ⋊ M16

Subgroups

subgroupG/H
centerC4×C9C2×D3
commutatorC2×C3C2×C4×C9
FrattiniC3×C4C2×C3×D3
FittingC3×C8×C9C2
socleC2×C32C2×C3×C4
prime radicalC2×C3×D3C3×C4
cosocleC32×C4C22×C3
hypercenterC8×C9D3
hypocenterC3C9×M16
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C4×C9 ◁ C8×C9

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 144

A minimal faithful permutation representation — 2 generators on 20 points:

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

Conjugacy classes & character table

162 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11720796, #11720797, D9×ΓL(1,𝔽25), …
maximal normal subgroup of#11720796, #11720797, D9×ΓL(1,𝔽25), …
maximal quotient ofC9×Dic3⋊C8, C9×#815, C9×(C2×D3)⋊C8, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 110);