Finite Group Databasebeta

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Cayley diagram

C22×C22⋊H(3,𝔽3) #88903

order432
isoclinismisoclinic class of C9⋊A4
automorphism?
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C22×C32

Aliases

direct productC22×C22⋊H(3,𝔽3)

Splittings

FittingC24×C32 ⋊ C3
SylowC24 ⋊ H(3,𝔽3)

Subgroups

subgroupG/H
centerC22×C3C3×A4
commutator, cosocleC22×C3C22×C32
FrattiniC3C22×C3×A4
FittingC24×C32C3
socleC24×C3C32
hypercenterC22×C32A4
hypocenterC22C22×H(3,𝔽3)
LCSG ▷ C22×C3 ▷ C22
UCSC1 ◁ C22×C3 ◁ C22×C32

224 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 36

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

(3,4)(5,8)(6,7)(9,10,13)(11,12,16)(14,15,17);(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×(C3⋊SL(2,ℤ/4))⋊C3, #64269⋊C3, C2×#88613, …
maximal normal subgroup ofC2×(C3⋊SL(2,ℤ/4))⋊C3, #64269⋊C3, C2×#88613, …
maximal quotient ofC2×C4×C22⋊H(3,𝔽3), D4×C22⋊H(3,𝔽3), C22×Q8⋊H(3,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 550);