Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | ((C3⋊S4)⋊C3)⋊C2 |
| commutator, hypocenter | Q8⋊H(3,𝔽3) | C2×C4 |
| Frattini, socle | C22×C3 | D3×S4 |
| Fitting | C2×C32×Q8 | C2×D3 |
| prime radical, cosocle | C2×Q8⋊H(3,𝔽3) | C22 |
| LCS | G ▷ Q8⋊H(3,𝔽3) | |
| UCS | C1 ◁ C22 | |
287 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 287 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 576 |
Polycyclic presentation:
2^3,3^3,2^3|a^2=c,b^2=i,g^2=i,h^2=i,e^a=e^2,f^a=f^2,d^b=d^2,f^b=f^2,g^b=gh,h^b=hi,e^d=ef,g^d=h,h^d=ghi,h^g=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 12392);