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C23×D3×3−1+2 #13049349

order1296
automorphism?
homologyAbelianization (H1) = C24×C32 · Schur multiplier (H2) = C26

Aliases

direct productC23×D3×3−1+2
unambiguous semidirect productC3⋊(C24×3−1+2)
unambiguous semidirect product(C2×C3)⋊(C23×3−1+2)

Splittings

FittingC23×C3×3−1+2 ⋊ C2
SylowC3×3−1+2 ⋊ C24

Subgroups

subgroupG/H
centerC23×C3C32×D3
commutator, cosocleC32C24×C32
FrattiniC3C23×C32×D3
FittingC23×C3×3−1+2C2
socleC23×C32C2×C32
prime radicalC23×C32×D3C3
hypercenterC23×3−1+2D3
hypocenterC3C24×3−1+2
LCSG ▷ C32 ▷ C3
UCSC1 ◁ C23×C3 ◁ C23×3−1+2

1248 conjugacy classes of subgroups; only the 581 normal subgroups are shown.

normal subgroups only

This group has 1248 conjugacy classes of subgroups — too many to draw. Only its 581 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

264 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC23×#92265, C23×D3×AGL(1,ℤ/9), 3−1+2×#937, …
maximal normal subgroup ofC23×D3×AGL(1,ℤ/9), 3−1+2×#937, #72220033, …
maximal quotient of3−1+2×#1004, C23×#92265, C2×D3×D4×3−1+2

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1296, 3067);