Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C28 |
| direct product | C3×#34403 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator | C22 | C24×C3×C4 |
| Frattini, cosocle | C23 | C25×C3 |
| prime radical | C24×C3 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1898 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 39 points:
(1,2,3)(4,5,6,7)(9,13)(10,16)(15,20)(17,22)(24,25)(26,33)(27,30)(28,36)(29,32)(31,37)(34,38)(35,39);(9,13)(14,19)(15,20)(21,23)(24,26,29,27)(25,30,32,33)(28,34,37,35)(31,38,36,39);(8,9)(10,14)(11,15)(12,13)(16,19)(17,21)(18,20)(22,23)(24,27,29,26)(25,30,32,33)(28,35,37,34)(31,38,36,39);(8,10,12,16)(9,14,13,19)(11,17,18,22)(15,21,20,23)(24,25)(26,33)(27,30)(28,31)(29,32)(34,39)(35,38)(36,37);(8,11,12,18)(9,15,13,20)(10,17,16,22)(14,21,19,23)(24,28,29,37)(25,31,32,36)(26,34,27,35)(30,38,33,39)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 27391);