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) = C22×C32 |
| direct product | C2×H(3,𝔽3)×#108 |
| unambiguous semidirect product | (C2×C32×#108)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C32×D4 |
| commutator | C3×C4 | C22×C32×C4 |
| Frattini, cosocle | C2×C3×C4 | C23×C32 |
| socle | C22×C3 | C32×C4⋊C4 |
| prime radical | C23×H(3,𝔽3) | Q8 |
| LCS | G ▷ C3×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C22×C4×H(3,𝔽3) ◁ G | |
583 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2)(19,20,22)(21,24,26)(23,27,25);(3,4,7,10,8,11,6,16)(5,12,14,17,15,18,13,9)(20,23,24)(22,26,27);(3,5,6,13,8,15,7,14)(4,9,10,12,11,17,16,18)308 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11109);