Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C25×C4 |
| direct product | C3×#10935 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | #120 |
| commutator | C23 | C2×C3×C42 |
| Frattini, cosocle | C25 | C23×C3 |
| prime radical | C26×C3 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C25×C3 ◁ G | |
2594 conjugacy classes of subgroups; only the 302 normal subgroups are shown.
This group has 2594 conjugacy classes of subgroups — too many to draw. Only its 302 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(5,7)(12,13)(14,18,19,16)(15,17);(8,9,10,11)(12,14)(13,16)(15,19)(17,18);(4,5,6,7)(12,13)(14,16)(15,17)(18,19)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3923);