Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #272 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C43 |
| direct product | C42×#1084 |
| derived, Sylow | H(3,𝔽3) ⋊ C43 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C42 | C3⋊D3 |
| commutator, hypocenter | H(3,𝔽3) | C43 |
| Frattini, socle | C23×C3 | C22×C3⋊D3 |
| Fitting | C2×C42×H(3,𝔽3) | C2 |
| prime radical, cosocle | C23×H(3,𝔽3) | C23 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C2×C3×C42 | |
1419 conjugacy classes of subgroups.
| transitive degree | μt(G) = 576 |
320 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 10316);