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C27⋊SL(2,𝔽3) #10619568

order648
automorphism?
homologyAbelianization (H1) = C3×C9 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC27⋊SL(2,𝔽3)

Splittings

FittingQ8×C27 ⋊ C3
SylowQ8 ⋊ C27⋊C3

Subgroups

subgroupG/H
center, FrattiniC2×C9C3×A4
commutatorC3×Q8C3×C9
FittingQ8×C27C3
socleC2×C3C9×A4
prime radicalC3×SL(2,𝔽3)C9
cosocleQ8×C9C32
hypercenterC2×C27A4
hypocenterQ8C27⋊C3
LCSG ▷ C3×Q8 ▷ Q8
UCSC1 ◁ C2×C9 ◁ C2×C27

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 216

Polycyclic presentation:

3^4,2^3|b^3=c,c^3=d,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

maximal subgroup of#13047028, D27⋊SL(2,𝔽3), C2×C27⋊SL(2,𝔽3), …
maximal normal subgroup of#13047028, D27⋊SL(2,𝔽3), C2×C27⋊SL(2,𝔽3), …
maximal quotient ofC2×C27⋊SL(2,𝔽3), (Q8⋊C9)⋊C27, #14515860, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(648, 87);