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(C3×SL(2,𝔽3))⋊C9 #10619577

order648
automorphism?
homologyAbelianization (H1) = C3×C9 · Schur multiplier (H2) = C32

Aliases

unambiguous semidirect product(C3×SL(2,𝔽3))⋊C9

Splittings

FittingC3×Q8×C9 ⋊ C3
SylowQ8 ⋊ C32⋊C9

Subgroups

subgroupG/H
center, Frattini, socleC2×C32C3×A4
commutatorC3×Q8C3×C9
FittingC3×Q8×C9C3
prime radicalC32×SL(2,𝔽3)C3
cosocleC32×Q8C32
hypercenterC2×C3×C9A4
hypocenterQ8C32⋊C9
LCSG ▷ C3×Q8 ▷ Q8
UCSC1 ◁ C2×C32 ◁ C2×C3×C9

79 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 216

Polycyclic presentation:

3^4,2^3|b^3=c,e^2=g,f^2=g,b^a=bd,e^a=f,f^a=efg,f^e=fg

Conjugacy classes & character table

93 conjugacy classes — view the full character table.

Relationships

derived subgroup of#13046995, #13046996
maximal subgroup of(C3⋊#226)⋊C9, (C3⋊GL(2,𝔽3))⋊C9, #13046995, …
maximal normal subgroup of(C3⋊#226)⋊C9, (C3⋊GL(2,𝔽3))⋊C9, #13046995, …
maximal quotient ofC2×(C3×SL(2,𝔽3))⋊C9, #14515838, #14515839, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(648, 96);