Finite Group Databasebeta

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Q8×#6180 #13144439

order1728
automorphism?
homologyAbelianization (H1) = C22×C8 · Schur multiplier (H2) = C22

Aliases

direct productQ8×#6180

Splittings

SylowH(3,𝔽3) ⋊ C8×Q8

Subgroups

subgroupG/H
center, FrattiniC2×C3×C4C22×C3⋊D3
commutatorC2×H(3,𝔽3)C22×C8
FittingC4×Q8×H(3,𝔽3)C2
socleC22×C3C22×C3⋊Dic3
prime radicalC22×H(3,𝔽3)C22×C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC3×C4×Q8C3⋊D3
hypocenterH(3,𝔽3)C8×Q8
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3×C4 ◁ C3×C4×Q8

561 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 561 conjugacy classes of subgroups — too many to draw. Only its 237 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

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 10502);