Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3)×H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C42 · Schur multiplier (H2) = C2×C32×C4 |
| direct product | H(3,𝔽3)×H(3,ℤ/4) |
| unambiguous semidirect product | (C32×H(3,ℤ/4))⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, commutator | C3×C4 | C32×C42 |
| Frattini, cosocle | C22×C3×C4 | C22×C32 |
| socle | C2×C3 | C32×#95 |
| prime radical | C23×H(3,𝔽3) | D4 |
| LCS | G ▷ C3×C4 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ G | |
517 conjugacy classes of subgroups; only the 189 normal subgroups are shown.
This group has 517 conjugacy classes of subgroups — too many to draw. Only its 189 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)(12,18,20,24)(13,16)(14,17)(15,19,22,25)(21,23);(2,5,6)(4,8,9)(10,12)(11,15,17,22)(13,19)(14,20)(16,18,23,24)(21,25)242 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2076);