Finite Group Databasebeta

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Cayley diagram

C22×C9⋊A4 #88900

order432
isoclinismisoclinic class of C9⋊A4
automorphismAut(G) = C3×D32×A4 (order 1296) · Inn(G) = C3×A4 · Out(G) = C2×C3×D3
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C22

Aliases

direct productC22×C9⋊A4

Splittings

FittingC24×C9 ⋊ C3
SylowC24 ⋊ 3−1+2

Subgroups

subgroupG/H
centerC22×C3C3×A4
commutator, cosocleC22×C3C22×C32
FrattiniC3C22×C3×A4
FittingC24×C9C3
socleC24×C3C32
prime radicalC22×C3×A4C3
hypercenterC22×C9A4
hypocenterC22C22×3−1+2
LCSG ▷ C22×C3 ▷ C22
UCSC1 ◁ C22×C3 ◁ C22×C9

137 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 108

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

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×Dic9⋊A4, #611⋊A4, C2×C4×C9⋊A4, …
maximal normal subgroup ofC2×Dic9⋊A4, #611⋊A4, C2×C4×C9⋊A4, …
maximal quotient ofC2×C4×C9⋊A4, D4×C9⋊A4, C22×C9⋊SL(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 547);