Finite Group DatabasebetaFinite Group Databasebeta| order | 972 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C4 · Schur multiplier (H2) = C33 |
| direct product | C3×C4×AΣL(1,𝔽27) |
| subgroup | G/H | |
|---|---|---|
| center | C32×C4 | H(3,𝔽3) |
| commutator | C32 | C33×C4 |
| Frattini, cosocle | C2×C32 | C2×C33 |
| socle | C2×C32 | C2×H(3,𝔽3) |
| prime radical | C2×C3×AΣL(1,𝔽27) | C2 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C4 ◁ C33×C4 ◁ G | |
378 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 378 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,3)(13,14,15,16);(4,5,7)(6,8,11)(9,12,10);(5,8,9)204 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(972, 394);