Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C2×C32 |
| unambiguous semidirect product | A4⋊#1084 |
| derived | A4×H(3,𝔽3) ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C3 | C3⋊(C3⋊S4) |
| commutator, hypocenter | A4×H(3,𝔽3) | C4 |
| Fitting | C23×H(3,𝔽3) | D3 |
| socle | C23×C3 | C3⋊(C3⋊D3) |
| prime radical, cosocle | C2×A4×H(3,𝔽3) | C2 |
| LCS | G ▷ A4×H(3,𝔽3) | |
| UCS | C1 ◁ C2×C3 | |
373 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 373 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
Polycyclic presentation:
2^2,3^4,2^2|a^2=b,c^a=c^2,d^a=d^2,e^a=e^2,g^a=gh,g^c=h,h^c=gh,e^d=ef62 conjugacy classes — view the full character table.
| maximal subgroup of | #72242699 |
| maximal quotient of | #72242699 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2137);