Finite Group Databasebeta

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C2×C8×C22⋊H(3,𝔽3) #13152213

order1728
isoclinismisoclinic class of C9⋊A4
automorphism?
homologyAbelianization (H1) = C2×C32×C8 · Schur multiplier (H2) = C22×C32

Aliases

direct productC2×C8×C22⋊H(3,𝔽3)
unambiguous semidirect product(C23×C8)⋊H(3,𝔽3)
unambiguous semidirect product(C2×C32×C8)⋊A4
unambiguous semidirect product(C2×C3×C8×A4)⋊C3

Splittings

FittingC23×C32×C8 ⋊ C3
SylowC23×C8 ⋊ H(3,𝔽3)

Subgroups

subgroupG/H
centerC2×C3×C8C3×A4
commutatorC22×C3C2×C32×C8
FrattiniC3×C4C22×C3×A4
FittingC23×C32×C8C3
socleC24×C3C32×C4
prime radicalC22×C22⋊H(3,𝔽3)C4
cosocleC22×C3×C4C22×C32
hypercenterC2×C32×C8A4
hypocenterC22C2×C8×H(3,𝔽3)
LCSG ▷ C22×C3 ▷ C22
UCSC1 ◁ C2×C3×C8 ◁ C2×C32×C8

536 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 144

Conjugacy classes & character table

320 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 18276);