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×#37676 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C2×C3×#546 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
3078 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)(9,11)(12,13)(14,20)(15,23)(16,17)(18,22)(19,24)(21,25)(26,27);(5,7)(13,17)(18,20)(19,23)(25,26);(4,5)(6,7)(8,9,10,11)(12,14)(13,18)(15,21)(16,22)(17,20)(19,25)(23,26)(24,27);(12,15,16,24)(13,19,17,23)(14,21,22,27)(18,25,20,26);(4,5)(6,7)(12,13)(14,18)(15,19)(16,17)(20,22)(21,25)(23,24)(26,27)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 30664);