Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7307 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C22 |
| direct product | C3×#7323 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #342 |
| commutator | C22×C4 | C3×C42 |
| Frattini, cosocle | C2×C4×Q8 | C22×C3 |
| prime radical | C23×C3 | #95 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×C4×Q8 ◁ G | |
256 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 256 conjugacy classes of subgroups — too many to draw. Only its 78 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)(4,5,6,9,7,8,10,11)(13,17,18,24)(19,27,26,25);(5,8)(6,10)(12,13,15,19,16,18,23,26)(14,17,20,25,21,24,22,27)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 311);