Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of H(3,𝔽3)×#453 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C23×C32 |
| direct product | H(3,𝔽3)×#453 |
| unambiguous semidirect product | (C32×#453)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C32×#120 |
| commutator, Frattini, cosocle | C2×C3×C4 | C23×C32 |
| LCS | G ▷ C2×C3×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C4×H(3,𝔽3) ◁ G | |
924 conjugacy classes of subgroups; only the 189 normal subgroups are shown.
This group has 924 conjugacy classes of subgroups — too many to draw. Only its 189 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2,4)(3,6,8)(5,9,7)(11,14)(12,16)(15,17);(2,5,6)(4,8,9)(10,11)(12,17)(13,15)(14,16);(12,16)(14,17)176 conjugacy classes — view the full character table.
| Fitting subgroup of | (C3⋊D3)⋊C3×#453 |
| maximal subgroup of | (C3⋊D3)⋊C3×#453 |
| maximal normal subgroup of | (C3⋊D3)⋊C3×#453 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11181);