Finite Group Databasebeta

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C2×#11722294 #13164306

order1728
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C25

Aliases

direct productC2×#11722294

Splittings

SylowH(3,𝔽3) ⋊ C2×#137

Subgroups

subgroupG/H
centerC22C2×#1088⋊C2
commutator, cosocle#1084C24
FrattiniC2×C3C22×GO+(4,𝔽2)
FittingC2×C4×H(3,𝔽3)D4
socleC22×C3C2×GO+(4,𝔽2)
prime radicalC2×#88650C2
hypercenterC2×C4#1088⋊C2
hypocenterH(3,𝔽3)C2×#137
LCSG ▷ #1084 ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

707 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 707 conjugacy classes of subgroups — too many to draw. Only its 97 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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 30369);