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C2×C3⋊(C3⋊#859) #13173068

order1728
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C33

Aliases

direct productC2×C3⋊(C3⋊#859)
unambiguous semidirect product(C2×C3)⋊(C3⋊#859)

Splittings

derivedC32×SL(2,𝔽3) ⋊ C2×C4

Subgroups

subgroupG/H
center, hypercenterC23C3⋊(C3⋊S4)
commutator, hypocenterC32×SL(2,𝔽3)C2×C4
FrattiniC22C2×C3⋊(C3⋊S4)
FittingC22×C32×Q8D3
socleC23×C32S4
prime radicalC22×C32×SL(2,𝔽3)C2
cosocleC2×C32×SL(2,𝔽3)C22
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C23

1142 conjugacy classes of subgroups; only the 239 normal subgroups are shown.

normal subgroups only

This group has 1142 conjugacy classes of subgroups — too many to draw. Only its 239 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degreeμt(G) = 576

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 39131);