Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×((C3⋊S4)⋊C3)⋊C2 |
| commutator, cosocle, hypocenter | Q8⋊H(3,𝔽3) | C23 |
| Frattini, socle | C2×C3 | C2×D3×S4 |
| Fitting | C2×C32×Q8 | C2×D3 |
| hypercenter | C22 | ((C3⋊S4)⋊C3)⋊C2 |
| LCS | G ▷ Q8⋊H(3,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C22 | |
570 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 570 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^3,3^3,2^3|b^2=i,g^2=i,h^2=i,c^a=ci,e^a=e^2,f^a=f^2,c^b=ci,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=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30261);