Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3)×H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C42 · Schur multiplier (H2) = C22 |
| direct product | 3−1+2×#338 |
| subgroup | G/H | |
|---|---|---|
| center, commutator | C3×C4 | C32×C42 |
| Frattini, cosocle | C22×C3×C4 | C22×C32 |
| socle | C2×C3 | C32×#95 |
| prime radical | C23×C32 | C3×Q8 |
| LCS | G ▷ C3×C4 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ G | |
344 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,4,3,6,8,7,5,9)(10,11,13,17,15,18,14,23)(12,19,21,16,22,24,20,25);(2,5,6)(4,8,9)(10,12,14,21,15,22,13,20)(11,16,17,19,18,25,23,24)242 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2080);