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C3⋊((C3⋊(C3⋊C8))⋊C3) #10619721

order648
isoclinismisoclinic class of C3⋊((C3⋊D3)⋊C3)
automorphism?
homologyAbelianization (H1) = C3×C8 · Schur multiplier (H2) = C32

Aliases

unambiguous semidirect productC3⋊((C3⋊(C3⋊C8))⋊C3)

Splittings

derivedC33 ⋊ C3×C8
SylowC3×H(3,𝔽3) ⋊ C8

Subgroups

subgroupG/H
center, hypercenterC4C3⋊((C3⋊D3)⋊C3)
commutator, hypocenterC33C3×C8
FrattiniC3×C4C3×C3⋊D3
FittingC3×C4×H(3,𝔽3)C2
socleC2×C32C3×Dic3
prime radicalC2×C3×H(3,𝔽3)C4
cosocleC33×C4C2×C3
LCSG ▷ C33
UCSC1 ◁ C4

149 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 216

Polycyclic presentation:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3⋊C8×(C3⋊D3)⋊C3, #88787⋊C3, D3×(C3⋊(C3⋊C8))⋊C3, …
maximal normal subgroup ofC3⋊C8×(C3⋊D3)⋊C3, #88787⋊C3, D3×(C3⋊(C3⋊C8))⋊C3, …
maximal quotient ofC3⋊((C3⋊(C3⋊C16))⋊C3), C2×C3⋊((C3⋊(C3⋊C8))⋊C3), C3⋊((C3⋊(C3⋊C8))⋊C9), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(648, 240);