Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#29647 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G | |
990 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3)(8,9,12,15)(10,16,18,13)(11,14,19,21)(17,22,23,20)(24,25,28,32)(26,29,33,37)(27,30,35,31)(34,36,39,38);(4,5)(6,7)(8,10,12,18)(9,13,15,16)(11,17,19,23)(14,20,21,22)(24,26)(25,29)(27,34)(28,33)(30,36)(31,38)(32,37)(35,39);(5,7)(8,11)(9,14)(10,17)(12,19)(13,20)(15,21)(16,22)(18,23)(25,30)(27,35)(29,36)(31,32)(34,39)(37,38);(11,19)(14,21)(17,23)(20,22)(26,33)(29,37)(34,39)(36,38)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 22635);