Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#38334 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1958 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)(5,6)(9,10)(12,13)(14,15)(16,17)(18,19);(4,5,8,6)(7,10,11,9)(13,17)(15,18)(21,24)(22,26);(12,14)(13,18)(15,17)(16,19)(20,21,23,24)(22,25,26,27);(4,6,8,5)(7,10,11,9)(12,14)(13,18)(15,17)(16,19);(4,7)(5,9)(6,10)(8,11)(12,15,16,18)(13,14,17,19)(20,22)(21,25)(23,26)(24,27)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 31322);