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C3⋊(C3⋊((C3⋊Dic3)⋊C3)) #11758018

order972
isoclinismisoclinic class of C3⋊(C3⋊((C3⋊D3)⋊C3))
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C34

Aliases

unambiguous semidirect productC3⋊(C3⋊((C3⋊Dic3)⋊C3))

Splittings

derivedC34 ⋊ C3×C4
SylowC32×H(3,𝔽3) ⋊ C4

Subgroups

subgroupG/H
center, hypercenterC2C3⋊(C3⋊((C3⋊D3)⋊C3))
commutator, hypocenterC34C3×C4
FrattiniC2×C3C3×C3⋊(C3⋊D3)
Fitting, prime radicalC2×C32×H(3,𝔽3)C2
socleC2×C33C3×D3
cosocleC2×C34C2×C3
LCSG ▷ C34
UCSC1 ◁ C2

758 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 324

Polycyclic presentation:

2,3,2,3^4|a^2=c,d^a=d^2,e^a=e^2,f^a=f^2,g^a=g^2,d^b=dg

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#14518665, #14518666, C3⋊Dic3×(C3⋊D3)⋊C3, …
maximal normal subgroup of#14518665, #14518666, C3⋊Dic3×(C3⋊D3)⋊C3, …
maximal quotient ofC3⋊(C3⋊((C3⋊(C3⋊C8))⋊C3)), C2×C3⋊(C3⋊((C3⋊Dic3)⋊C3)), #72228915

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(972, 743);