Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13343 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#13375 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C23×C3 |
| prime radical | C2×C3×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3 ◁ C22×C3×D4 ◁ G | |
704 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 704 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 192 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(4,5,7,9)(6,10,11,8)(13,16)(17,19);(4,6,7,11)(5,8,9,10)(14,18);(12,13)(14,16)(15,17)(18,19)93 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6363);