Finite Group Databasebeta

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C2×#11720529 #13144343

order1728
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23

Aliases

direct productC2×#11720529

Splittings

SylowH(3,𝔽3) ⋊ C2×#107

Subgroups

subgroupG/H
center, socleC23×C3#638
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C22×C3⋊D3
FittingC2×C4⋊C4×H(3,𝔽3)C2
prime radicalC23×H(3,𝔽3)D4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C3×C4⋊C4C3⋊D3
hypocenterH(3,𝔽3)C2×#107
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C2×C3×C4⋊C4

715 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 715 conjugacy classes of subgroups — too many to draw. Only its 229 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

164 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 10406);