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 | C2 | #11720654 |
| commutator, cosocle | #41⋊C27 | C22 |
| Frattini | C4×C9 | C2×S4 |
| Fitting | C9×H(5,𝔽2) | D3 |
| socle | C2×C3 | #63694 |
| hypercenter | Q8 | (C22⋊C27)⋊C2 |
| hypocenter | Q8⋊C27 | D4 |
| LCS | G ▷ #41⋊C27 ▷ Q8⋊C27 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q8 | |
198 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 27 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
Polycyclic presentation:
2^3,3^3,2^3|b^2=i,c^2=i,d^3=e,e^3=f,g^2=i,h^2=i,b^a=bc,c^a=ci,d^a=d^2e^2f^2,e^a=e^2f^2,f^a=f^2,g^a=gh,h^a=hi,c^b=ci,g^d=h,h^d=ghi,h^g=hi93 conjugacy classes — view the full character table.
| maximal subgroup of | #72276227, #72276259 |
| maximal normal subgroup of | #72276259 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2230);