Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25×C4 |
| direct product | C3×#25761 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C4×D4 ◁ G | |
1582 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)(5,6)(9,11)(12,13)(14,21)(15,23)(16,19)(17,26)(18,22)(20,25)(24,27);(4,5,7,6)(12,14,16,22)(13,18,19,21)(15,17,24,25)(20,27,26,23);(8,9,10,11)(12,15,17,22,16,24,25,14)(13,18,20,23,19,21,26,27);(4,6,7,5)(13,19)(18,21)(20,26)(23,27)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 18749);