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×#37682 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
3174 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)(14,19)(15,23)(16,21)(17,18)(20,25)(22,24)(26,27);(5,7)(13,18)(14,21)(20,23)(22,27);(4,5)(6,7)(8,9,10,11)(12,14,17,21)(13,19,18,16)(15,22,25,27)(20,26,23,24);(12,15,17,25)(13,20,18,23)(14,22,21,27)(16,24,19,26);(4,5)(6,7)(12,16,17,19)(13,21,18,14)(15,24,25,26)(20,27,23,22)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 30670);