Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of H(3,𝔽3)×B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C24 |
| direct product | 3−1+2×B(4,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C32×#120 |
| commutator, Frattini, cosocle | C23×C3 | C23×C32 |
| prime radical | C32×B(4,𝔽2) | C3 |
| LCS | G ▷ C23×C3 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C23×3−1+2 ◁ G | |
792 conjugacy classes of subgroups; only the 189 normal subgroups are shown.
This group has 792 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:
(1,2,4,3,6,8,7,5,9)(14,17);(2,5,6)(4,8,9)(10,11)(12,14)(13,15)(16,17);(10,12)(11,14)(13,16)(15,17)176 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,ℤ/9)×B(4,𝔽2), 3−1+2×#5032, 3−1+2×#5934, … |
| maximal normal subgroup of | AGL(1,ℤ/9)×B(4,𝔽2), C2×3−1+2×B(4,𝔽2) |
| maximal quotient of | 3−1+2×#5032, C2×3−1+2×B(4,𝔽2) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11194);