Finite Group Databasebeta

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C22×C4⋊#1088 #13164669

order1728
isoclinismisoclinic class of #88627
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C26

Aliases

direct productC22×C4⋊#1088

Splittings

FittingC22×C4×H(3,𝔽3) ⋊ C4
SylowH(3,𝔽3) ⋊ C22×C4⋊C4

Subgroups

subgroupG/H
center, socleC23×C3C2×#157
commutatorC2×H(3,𝔽3)C23×C4
FrattiniC2×C3C23×#157
FittingC22×C4×H(3,𝔽3)C4
prime radicalC23×#272C22
cosocleC2×#272C24
hypercenterC22×C3×C4#157
hypocenterH(3,𝔽3)C22×C4⋊C4
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

1099 conjugacy classes of subgroups; only the 223 normal subgroups are shown.

normal subgroups only

This group has 1099 conjugacy classes of subgroups — too many to draw. Only its 223 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

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 30732);