Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C2×C3×#2052 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C23×C3×C4 ◁ G | |
1362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2)(23,24,25);(3,4)(5,6)(7,8)(9,17)(10,11)(12,15)(13,16)(14,19)(18,21)(20,22);(7,9,13,21)(8,14,16,22)(10,17,15,18)(11,20,12,19);(4,6)(7,10,11,8,13,15,12,16)(9,18,19,14,21,17,20,22)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 16517);