Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8×H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×3−1+2×#107 |
| unambiguous semidirect product | (C2×C9×#107)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C32×D4 |
| commutator | C3×C4 | C22×C32×C4 |
| Frattini, cosocle | C2×C3×C4 | C23×C32 |
| prime radical | C23×C32 | C3×D4 |
| LCS | G ▷ C3×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C22×C4×3−1+2 ◁ G | |
520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2)(15,16,18,17,20,22,21,19,23);(3,4,5,6)(8,11)(9,12)(13,14)(16,19,20)(18,22,23);(7,8,12,14,10,13,9,11)308 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11101);