Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3)×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C26 |
| direct product | 3−1+2×#379 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23×C3 | C23×C32 |
| commutator | C22×C3 | C22×C32×C4 |
| prime radical | C25×C32 | C2×C3 |
| LCS | G ▷ C22×C3 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2024 conjugacy classes of subgroups; only the 455 normal subgroups are shown.
This group has 2024 conjugacy classes of subgroups — too many to draw. Only its 455 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,4,3,6,8,7,5,9)(11,13)(14,15,16,17)(19,21);(2,5,6)(4,8,9)(18,19)(20,21);(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)308 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 10960);