Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C23×H(3,𝔽3) ⋊ D4 |
| Sylow | H(3,𝔽3) ⋊ #392 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×#1088⋊C2 |
| commutator, cosocle | C22×#272 | C23 |
| Frattini, socle | C22×C3 | C2×GO+(4,𝔽2) |
| Fitting | C23×H(3,𝔽3) | D4 |
| hypercenter | C23 | #1088⋊C2 |
| hypocenter | H(3,𝔽3) | #392 |
| LCS | G ▷ C22×#272 ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C23 | |
955 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 955 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^6,3^3|b^a=bd,c^a=cf,g^a=g^2,h^a=g^2h,i^a=i^2,c^b=ce,g^b=g^2h^2,h^b=hi,i^b=i^2,g^d=g^2i^2,h^d=h^2i^2,h^g=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 12771);