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Cayley diagram

C2×C3⋊((C22⋊C9)⋊C2) #88890

order432
isoclinismisoclinic class of C3⋊((C22⋊C9)⋊C2)
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C22×C3

Aliases

direct productC2×C3⋊((C22⋊C9)⋊C2)
unambiguous semidirect product(C2×C3)⋊((C22⋊C9)⋊C2)

Splittings

derivedC3×C22⋊C9 ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC2C3⋊((C22⋊C9)⋊C2)
commutator, cosocle, hypocenterC3×C22⋊C9C22
FrattiniC3C2×C3⋊S4
Fitting, socleC23×C32D3
LCSG ▷ C3×C22⋊C9
UCSC1 ◁ C2

178 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 54

A minimal faithful permutation representation — 3 generators on 18 points:

(1,2)(4,5)(6,7,9,8)(11,13)(12,16)(14,15)(17,18);(1,2)(3,4,5);(7,8,9)(10,18,13,16,15,14,12,11,17)

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C3⋊((C22⋊C9)⋊C2))⋊C4, #11722264, #11722267, …
maximal normal subgroup of(C3⋊((C22⋊C9)⋊C2))⋊C4, #11722264, #11722267, …
maximal quotient of#11722377, C4×C3⋊((C22⋊C9)⋊C2), C4⋊(C3⋊((C22⋊C9)⋊C2)), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 537);