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C2×2−1+4⋊H(3,𝔽3) #13164862

order1728
automorphism?
homologyAbelianization (H1) = C23×C32 · Schur multiplier (H2) = C23×C32

Aliases

direct productC2×2−1+4⋊H(3,𝔽3)
unambiguous semidirect product(C2×2−1+4)⋊H(3,𝔽3)
unambiguous semidirect product(C2×C3×2−1+4⋊C3)⋊C3

Splittings

FittingC2×C32×2−1+4 ⋊ C3
SylowC2×2−1+4 ⋊ H(3,𝔽3)

Subgroups

subgroupG/H
center, socleC22×C3C22×C3×A4
commutator, cosocleC3×Q8C23×C32
FrattiniC2×C3C23×C3×A4
FittingC2×C32×2−1+4C3
hypercenterC2×C32×D4A4
hypocenterQ8C23×H(3,𝔽3)
LCSG ▷ C3×Q8 ▷ Q8
UCSC1 ◁ C22×C3 ◁ C2×C32×D4

1181 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 1181 conjugacy classes of subgroups — too many to draw. Only its 169 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) = 288

Conjugacy classes & character table

182 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 30925);