Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×C3×#2929 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C8 | C23 |
| commutator | C22 | C24×C3×C4 |
| Frattini, cosocle | C2×C4 | C25×C3 |
| socle | C23×C3 | C23×C4 |
| prime radical | C2×C3×#525 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C8 ◁ G | |
3046 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 25 points:
(1,2)(23,24,25);(3,4)(5,6)(7,8,10,14,11,13,17,20)(9,12,15,19,16,18,21,22);(4,6);(7,9,10,15,11,16,17,21)(8,12,14,19,13,18,20,22);(8,13)(12,18)(14,20)(19,22)336 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53121);