Finite Group Databasebeta

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Cayley diagram

C22×C9×A4 #88899

order432
isoclinismisoclinic class of A4
automorphism?
homologyAbelianization (H1) = C22×C3×C9 · Schur multiplier (H2) = C22×C3

Aliases

direct productC22×C9×A4

Splittings

FittingC24×C9 ⋊ C3
derivedC22 ⋊ C22×C3×C9
SylowC24 ⋊ C3×C9

Subgroups

subgroupG/H
center, hypercenterC22×C9A4
commutator, hypocenterC22C22×C3×C9
FrattiniC3C22×C3×A4
FittingC24×C9C3
socleC24×C3C32
prime radicalC22×C3×A4C3
cosocleC22×C3C22×C32
LCSG ▷ C22
UCSC1 ◁ C22×C9

157 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)(6,7,8);(1,2)(5,8)(6,7)(9,13,17,11,16,12,14,10,15)

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

derived subgroup ofA4×C9⋊S4, #72269058
maximal subgroup ofC2×C9×SL(2,ℤ/4), C9×GL(2,ℤ/4), C2×C9⋊SL(2,ℤ/4), …
maximal normal subgroup ofC2×C9×SL(2,ℤ/4), C9×GL(2,ℤ/4), C2×C9⋊SL(2,ℤ/4), …
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, 546);