Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C3×#14658 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C24 |
| commutator | C23 | C23×C3×C4 |
| Frattini, cosocle | C24 | C24×C3 |
| prime radical | C23×C3×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
1898 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(13,16)(17,19)(20,21,23,24)(22,25,27,26);(5,7)(8,9,10,11)(13,16)(14,18)(21,24)(25,26);(4,5)(6,7)(8,9)(10,11)(12,13)(14,17)(15,16)(18,19)(20,21)(22,26)(23,24)(25,27);(12,14)(13,17)(15,18)(16,19)(20,22)(21,25)(23,27)(24,26)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 7646);