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#10619582

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

Aliases

No aliases for this group.

Splittings

FittingQ8×3−1+2 ⋊ C3
SylowQ8 ⋊ #731

Subgroups

subgroupG/H
center, socleC2×C3C22⋊H(3,𝔽3)
commutator, cosocleC32×Q8C32
FrattiniC2×C32C3×A4
FittingQ8×3−1+2C3
prime radicalQ8⋊H(3,𝔽3)C3
hypercenterC2×3−1+2A4
hypocenterQ8#731
LCSG ▷ C32×Q8 ▷ C3×Q8 ▷ Q8
UCSC1 ◁ C2×C3 ◁ C2×C32 ◁ C2×3−1+2

50 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 216

Polycyclic presentation:

3^4,2^3|b^3=d^2,e^2=g,f^2=g,b^a=bc,c^a=cd,e^a=f,f^a=efg,c^b=cd^2,f^e=fg

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

derived subgroup of#13047004, #13047007
maximal subgroup of#13047004, #13047007, C2×#10619582, …
maximal normal subgroup of#13047004, #13047007, C2×#10619582, …
maximal quotient ofC2×#10619582, #14515834, Q8⋊#6862, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(648, 101);