Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1304 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C8 · Schur multiplier (H2) = C2×C4 |
| direct product | C3×#7034 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | #95 |
| commutator | C23 | C3×C4×C8 |
| Frattini, cosocle | C22×C42 | C22×C3 |
| prime radical | C24×C3 | C42 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G | |
406 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 406 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3)(4,5)(6,10,11,8)(7,9)(13,16)(14,18)(20,21,22,24,23,25,26,27);(4,6)(5,8)(7,11)(9,10)(12,13,14,17,15,16,18,19)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 22);