Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#39373 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C22×C3×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1954 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 31 points:
(1,2,3)(10,14)(12,15)(16,17,21,18)(19,24,27,22)(20,23,28,25)(26,30,31,29);(5,6)(8,9,11,13)(10,12,14,15)(17,18)(22,24)(23,25)(29,30);(4,5,7,6)(8,10,11,14)(9,12,13,15)(16,18,21,17)(19,24,27,22)(20,25,28,23)(26,30,31,29);(16,19,21,27)(17,22,18,24)(20,26,28,31)(23,29,25,30);(4,6,7,5)(16,20,21,28)(17,23,18,25)(19,26,27,31)(22,29,24,30)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 32361);