Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D32⋊S4 |
| commutator, cosocle | C3⋊(C3⋊(#41⋊C3)) | C22 |
| Fitting | C32×Q8 | #67 |
| socle | C2×C32 | GL(2,ℤ/4) |
| hypocenter | C32×SL(2,𝔽3) | D4 |
| LCS | G ▷ C3⋊(C3⋊(#41⋊C3)) ▷ C32×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
268 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 17 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^3,3^3,2^3|b^2=i,c^2=i,g^2=i,h^2=i,b^a=bci,c^a=ci,d^a=d^2,e^a=e^2,f^a=e^2f,g^a=gh,h^a=hi,c^b=ci,e^b=e^2f^2,e^c=e^2,f^c=f^2,g^d=h,h^d=ghi,h^g=hi39 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46272);