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×#37659 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
3110 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)(12,13,17,16)(14,20,22,18)(15,23,24,19)(21,25,27,26);(5,7)(12,14,17,22)(13,18,16,20)(15,21,24,27)(19,25,23,26);(4,5)(6,7)(8,9)(10,11)(12,14)(13,18)(15,21)(16,20)(17,22)(19,25)(23,26)(24,27);(12,15,17,24)(13,19,16,23)(14,21,22,27)(18,25,20,26);(4,5)(6,7)(12,16,17,13)(14,20,22,18)(15,23,24,19)(21,26,27,25)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 30647);