Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C33 · Schur multiplier (H2) = C25×C33 |
| direct product | C2×C33×AGL(1,ℤ/8) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C33 | C2×D4 |
| commutator, Frattini, cosocle | C4 | C24×C33 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C33 ◁ C22×C33×C4 ◁ G | |
4172 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 432 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2)(11,12,13);(3,4)(5,7)(6,8)(9,10)(14,15,16);(4,7)(5,10)(8,9)(17,18,19);(4,8)(7,9)594 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×C3×C3⋊D3×AGL(1,ℤ/8) |
| maximal subgroup of | C2×C3×C3⋊D3×AGL(1,ℤ/8) |
| maximal normal subgroup of | C2×C3×C3⋊D3×AGL(1,ℤ/8) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46091);