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C3×C4×((C3⋊D3)⋊C3)⋊C2 #13048552

order1296
isoclinismisoclinic class of ((C3⋊D3)⋊C3)⋊C2
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23

Aliases

direct productC3×C4×((C3⋊D3)⋊C3)⋊C2

Splittings

FittingC3×C4×H(3,𝔽3) ⋊ C22
derivedH(3,𝔽3) ⋊ C22×C3×C4
SylowC3×H(3,𝔽3) ⋊ C22×C4

Subgroups

subgroupG/H
center, hypercenterC3×C4((C3⋊D3)⋊C3)⋊C2
commutator, hypocenterH(3,𝔽3)C22×C3×C4
FrattiniC2×C3C2×C3×D32
FittingC3×C4×H(3,𝔽3)C22
socleC2×C32C2×D32
prime radicalC2×C3×((C3⋊D3)⋊C3)⋊C2C2
cosocleC2×H(3,𝔽3)C23×C3
LCSG ▷ H(3,𝔽3)
UCSC1 ◁ C3×C4

507 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 507 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 72

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×C4×AGL(2,𝔽3), C3×AGL(1,𝔽5)×((C3⋊D3)⋊C3)⋊C2, C2×C3×C4×((C3⋊D3)⋊C3)⋊C2, …
maximal normal subgroup ofC2×C3×C4×((C3⋊D3)⋊C3)⋊C2, C3×C8×((C3⋊D3)⋊C3)⋊C2, C3×C4×C5×((C3⋊D3)⋊C3)⋊C2, …
maximal quotient ofC3×AGL(1,𝔽5)×((C3⋊D3)⋊C3)⋊C2, C2×C3×C4×((C3⋊D3)⋊C3)⋊C2, C3×C8×((C3⋊D3)⋊C3)⋊C2, …
automorphism group ofC5×AΣL(1,𝔽27), C5×#731

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1296, 2270);