Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of H(3,𝔽3)×#353 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C2×C32 |
| direct product | H(3,𝔽3)×#355 |
| unambiguous semidirect product | (C32×#355)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C32×#99 |
| commutator | C2×C3×C4 | C2×C32×C4 |
| Frattini, cosocle | C2×C3×Q8 | C22×C32 |
| prime radical | C2×D4×H(3,𝔽3) | C4 |
| LCS | G ▷ C2×C3×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×H(3,𝔽3) ◁ C2×Q8×H(3,𝔽3) ◁ G | |
352 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 352 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,4)(3,6,8)(5,9,7)(10,11,13,17,14,18,22,25)(12,16,20,24,19,15,21,23);(2,5,6)(4,8,9)(11,15)(12,19)(16,18)(17,24)(20,21)(23,25)143 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2130);