Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C16 · Schur multiplier (H2) = C1 |
| direct product | 3−1+2×M64 |
| unambiguous semidirect product | (C9×M64)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C3×C16 | C22×C32 |
| commutator, socle | C2×C3 | C2×C32×C16 |
| prime radical | C22×C32 | C3×C16 |
| LCS | G ▷ C2×C3 ▷ C1 | |
| UCS | C1 ◁ C3×C16 ◁ G | |
128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 2 generators on 41 points:
(1,2,4,3,6,8,7,5,9)(10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41);(2,5,6)(4,8,9)(11,27)(13,29)(15,31)(17,33)(19,35)(21,37)(23,39)(25,41)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2176);