Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of H(3,𝔽3)×#99 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×3−1+2×#100 |
| unambiguous semidirect product | (C2×C9×#100)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C32×C22⋊C4 |
| commutator | C22×C3 | C22×C32×C4 |
| Frattini, cosocle | C2×C3×C4 | C23×C32 |
| prime radical | C22×C32×D4 | C2×C3 |
| LCS | G ▷ C22×C3 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C4×3−1+2 ◁ G | |
744 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2)(11,12,14,13,16,18,17,15,19);(3,4,5,8,6,7,9,10)(12,15,16)(14,18,19);(4,7)(5,9)242 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×#100×AGL(1,ℤ/9), C2×#10588427⋊C3 |
| maximal subgroup of | C2×#100×AGL(1,ℤ/9), C2×3−1+2×#453, C2×#10588427⋊C3 |
| maximal normal subgroup of | C2×#100×AGL(1,ℤ/9), C2×3−1+2×#453, C2×#10588427⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11056);