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Dic9⋊SL(2,𝔽3) #11720750

order864
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productDic9⋊SL(2,𝔽3)
unambiguous semidirect product(C9⋊SL(2,𝔽3))⋊C4

Splittings

derivedQ8×C9 ⋊ C3×C4

Subgroups

subgroupG/H
center, hypercenterC22D9⋊A4
commutator, hypocenterQ8×C9C3×C4
Frattini, socleC22×C3D3×A4
Fitting, cosocleC2×Q8×C9C2×C3
prime radicalC2×C3×SL(2,𝔽3)D3
LCSG ▷ Q8×C9
UCSC1 ◁ C22

97 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 288

Polycyclic presentation:

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

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic18⋊SL(2,𝔽3), C4×#88614, #88614⋊C4, …
maximal normal subgroup ofDic18⋊SL(2,𝔽3), C4×#88614, #88614⋊C4, …
maximal quotient of(C9⋊C8)⋊SL(2,𝔽3), C2×Dic9⋊SL(2,𝔽3)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(864, 717);