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) = C23×C32 |
| direct product | H(3,𝔽3)×#452 |
| unambiguous semidirect product | (C32×#452)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×C32×D4 |
| commutator, Frattini, cosocle | C2×C3×C4 | C23×C32 |
| prime radical | C2×#41×H(3,𝔽3) | C2 |
| LCS | G ▷ C2×C3×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C4×H(3,𝔽3) ◁ G | |
836 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,4)(3,6,8)(5,9,7)(10,11)(12,13)(14,15,19,22)(16,23,25,28)(17,21,18,26)(20,27,29,24);(2,5,6)(4,8,9)(14,16)(15,20)(17,24)(18,27)(19,25)(21,28)(22,29)(23,26);(11,13)(14,17,19,18)(15,21,22,26)(16,24,25,27)(20,28,29,23)209 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11178);