Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of H(3,𝔽3)×#447 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C2×C32 |
| direct product | H(3,𝔽3)×#478 |
| unambiguous semidirect product | (C32×#478)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×C32×D4 |
| commutator, Frattini, cosocle | C2×C3×C4 | C23×C32 |
| prime radical | C2×D4×H(3,𝔽3) | C22 |
| LCS | G ▷ C2×C3×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C42×H(3,𝔽3) ◁ G | |
561 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,4)(3,6,8)(5,9,7)(10,11,13,15)(12,16,17,14)(18,19,20,22)(21,23,25,24);(2,5,6)(4,8,9)(10,12,13,17)(11,14,15,16)(19,22)(23,24);(19,23)(21,25)(22,24)209 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11256);