Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C3×#37680 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
3002 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3)(9,11)(12,13)(14,20)(15,19)(16,17)(18,22)(21,26)(23,24)(25,27);(5,7)(13,17)(18,20)(19,24)(25,26);(4,5)(6,7)(8,9,10,11)(12,13)(14,18)(15,19)(16,17)(20,22)(21,25)(23,24)(26,27);(12,14,16,22)(13,18,17,20)(15,21,23,27)(19,25,24,26);(4,5)(6,7)(12,15,16,23)(13,19,17,24)(14,21,22,27)(18,25,20,26)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 30668);