Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of H(3,𝔽3)×AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C22 |
| direct product | 3−1+2×#430 |
| unambiguous semidirect product | (C9×#430)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C2×C32×D4 |
| commutator | C3×C4 | C22×C32×C4 |
| Frattini, cosocle | C2×C3×C4 | C23×C32 |
| socle | C2×C3 | C2×C32×C4⋊C4 |
| prime radical | C23×C32 | C3×Q8 |
| LCS | G ▷ C3×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C4×3−1+2 ◁ G | |
408 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,4,3,6,8,7,5,9)(10,11,14,17,15,18,13,23)(12,19,22,24,20,25,21,16);(2,5,6)(4,8,9)(10,12,13,21,15,20,14,22)(11,16,17,19,18,24,23,25);(12,20)(16,24)(19,25)(21,22)242 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11113);