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C22×((C22⋊C9)⋊C2)⋊C3 #11724043

order864
isoclinismisoclinic class of ((C22⋊C9)⋊C2)⋊C3
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C24

Aliases

direct productC22×((C22⋊C9)⋊C2)⋊C3
unambiguous semidirect product((C22⋊C9)⋊C2)⋊(C22×C3)
unambiguous semidirect product#1094⋊C23
unambiguous semidirect product(C2×#1094)⋊C22

Splittings

derivedC22⋊C9 ⋊ C23×C3

Subgroups

subgroupG/H
center, hypercenterC22((C22⋊C9)⋊C2)⋊C3
commutator, cosocle, hypocenterC22⋊C9C23×C3
FrattiniC3C22×C3×S4
FittingC24×C32D3
socleC24×C3C3×D3
LCSG ▷ C22⋊C9
UCSC1 ◁ C22

489 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 489 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 36

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C2×((C22⋊C9)⋊C2)⋊C3)⋊C4, C2×C4×((C22⋊C9)⋊C2)⋊C3, C2×C4⋊(((C22⋊C9)⋊C2)⋊C3), …
maximal normal subgroup of(C2×((C22⋊C9)⋊C2)⋊C3)⋊C4, C2×C4×((C22⋊C9)⋊C2)⋊C3, C2×C4⋊(((C22⋊C9)⋊C2)⋊C3), …
maximal quotient ofC2×#1094⋊Q8, C2×C4×((C22⋊C9)⋊C2)⋊C3, C2×C4⋊(((C22⋊C9)⋊C2)⋊C3), …
automorphism group of(C22⋊C9)⋊C8, #63421, #63422, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(864, 4010);