Finite Group Databasebeta

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Cayley diagram

C22×(C3⋊Dic3)⋊C3 #88729

order432
isoclinismisoclinic class of (C3⋊D3)⋊C3
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23×C3

Aliases

direct productC22×(C3⋊Dic3)⋊C3

Splittings

derivedC32 ⋊ C22×C3×C4
SylowH(3,𝔽3) ⋊ C22×C4

Subgroups

subgroupG/H
center, hypercenterC23(C3⋊D3)⋊C3
commutator, hypocenterC32C22×C3×C4
FrattiniC2×C3C22×C3×D3
Fitting, prime radicalC23×H(3,𝔽3)C2
socleC23×C3C3×D3
cosocleC2×C32C23×C3
LCSG ▷ C32
UCSC1 ◁ C23

221 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 144

A minimal faithful permutation representation — 3 generators on 17 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11720334, #63658⋊C3, #11721293, …
maximal normal subgroup of#11720334, #63658⋊C3, #11721293, …
maximal quotient ofC22×(C3⋊(C3⋊C8))⋊C3, C2×#3748⋊C3, C2×C4×(C3⋊Dic3)⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 376);