Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C42 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×C3×#2274 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C24 |
| commutator | C2 | C23×C3×C42 |
| Frattini, cosocle | C23 | C25×C3 |
| prime radical | C25×C3 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
4110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 21 points:
(1,2)(19,20,21);(7,8,10,9)(11,12,15,13)(14,17,18,16);(3,4,5,6)(7,9,10,8)(12,13)(16,17);(11,13,15,12)(14,17,18,16);(11,14)(12,16)(13,17)(15,18)408 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26983);