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C2×C3×C4×(C3⋊D3)⋊C3 #13048831

order1296
isoclinismisoclinic class of (C3⋊D3)⋊C3
automorphism?
homologyAbelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C23×C32

Aliases

direct productC2×C3×C4×(C3⋊D3)⋊C3
unambiguous semidirect product(C2×C4×C3⋊D3)⋊C32
unambiguous semidirect product(C2×C3×C4×C3⋊D3)⋊C3

Splittings

FittingC2×C3×C4×H(3,𝔽3) ⋊ C2
derivedC32 ⋊ C22×C32×C4
SylowC3×H(3,𝔽3) ⋊ C22×C4

Subgroups

subgroupG/H
center, hypercenterC2×C3×C4(C3⋊D3)⋊C3
commutator, hypocenterC32C22×C32×C4
FrattiniC2×C3C22×C32×D3
FittingC2×C3×C4×H(3,𝔽3)C2
socleC22×C32C2×C3×D3
prime radicalC22×C3×(C3⋊D3)⋊C3C2
cosocleC2×C32C23×C32
LCSG ▷ C32
UCSC1 ◁ C2×C3×C4

684 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 226 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) = 144

Conjugacy classes & character table

240 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×C3×AGL(1,𝔽5)×(C3⋊D3)⋊C3, C2×C3×C4×PSU(3,𝔽2)⋊C3, C2×C3×C4×((C3⋊D3)⋊C3)⋊C2, …
maximal normal subgroup ofC2×C3×C4×((C3⋊D3)⋊C3)⋊C2, C2×C3×D4×(C3⋊D3)⋊C3
maximal quotient ofC2×C3×AGL(1,𝔽5)×(C3⋊D3)⋊C3

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1296, 2549);