Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#35942 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C2×C3×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1994 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3)(12,13)(14,19)(15,17)(16,18)(20,21,24,27)(22,23,26,25);(4,5)(6,9)(7,10)(8,11)(12,13)(14,19)(15,17)(16,18)(20,22,24,26)(21,25,27,23);(4,6)(5,8)(7,11)(9,10)(12,14,16,15)(13,17,18,19)(20,21)(22,25)(23,26)(24,27);(4,6)(5,9)(7,11)(8,10)(12,15,16,14)(13,17,18,19)(20,23,24,25)(21,26,27,22);(4,5)(6,8)(7,10)(9,11)(12,13)(14,17)(15,19)(16,18)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 28930);