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Cayley diagram

C3×Dic9⋊C4 #88482

order432
isoclinismisoclinic class of Dic18
automorphismAut(G) = C25×AGL(1,ℤ/9) (order 1728) · Inn(G) = C2×D9 · Out(G) = C24×C3
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC3×Dic9⋊C4

Splittings

SylowC3×C9 ⋊ C4⋊C4

Subgroups

subgroupG/H
centerC22×C3C2×D9
commutatorC2×C9C2×C3×C4
FrattiniC22×C3C2×C3×D3
FittingC2×C3×C4×C9C2
socle, prime radicalC22×C32C2×D3
cosocleC22×C9C22×C3
hypercenterC2×C3×C4D9
hypocenterC9C3×C4⋊C4
LCSG ▷ C2×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

94 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)(4,9,11,10,5,7,6,12,8)(13,16,15,14)(18,20);(5,7)(6,10)(8,9)(11,12)(17,20,19,18)

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11720898, #11720900, #11720902, …
maximal normal subgroup of#11720898, #11720900, #11720902, …
maximal quotient ofC3×#63366, C3×#63367, C3×Dic9⋊C8, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 129);