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C2×C3×C22⋊H(3,𝔽3) #10620102

order648
isoclinismisoclinic class of C9⋊A4
automorphism?
homologyAbelianization (H1) = C2×C33 · Schur multiplier (H2) = C2×C34

Aliases

direct productC2×C3×C22⋊H(3,𝔽3)
unambiguous semidirect product(C23×C3)⋊H(3,𝔽3)

Splittings

FittingC23×C33 ⋊ C3
SylowC23 ⋊ C3×H(3,𝔽3)

Subgroups

subgroupG/H
centerC2×C32C3×A4
commutator, cosocleC22×C3C2×C33
FrattiniC3C2×C32×A4
FittingC23×C33C3
socleC23×C32C32
hypercenterC2×C33A4
hypocenterC22C2×C3×H(3,𝔽3)
LCSG ▷ C22×C3 ▷ C22
UCSC1 ◁ C2×C32 ◁ C2×C33

336 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 54

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×(C3⋊SL(2,ℤ/4))⋊C3, C3×#88613, C3⋊((C3⋊SL(2,ℤ/4))⋊C3), …
maximal normal subgroup ofC3×(C3⋊SL(2,ℤ/4))⋊C3, C3×#88613, C3⋊((C3⋊SL(2,ℤ/4))⋊C3), …
maximal quotient ofC3×C4×C22⋊H(3,𝔽3), C2×C3×Q8⋊H(3,𝔽3), C3×#41⋊H(3,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(648, 621);