Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60367 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C3×#60427 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C2×C3×#575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×#117 ◁ G | |
2678 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 39 points:
(1,2,3)(5,7)(8,9,14,19)(10,15,21,29)(11,24,26,33)(12,18,13,17)(16,27,32,25)(20,34,35,39)(22,31,23,30)(28,37,38,36);(4,5,6,7)(8,10)(9,15)(11,20)(12,22)(13,23)(14,21)(16,28)(17,30)(18,31)(19,29)(24,34)(25,36)(26,35)(27,37)(32,38)(33,39);(8,11)(9,16)(10,20)(12,25)(13,27)(14,26)(15,28)(17,24)(18,33)(19,32)(21,35)(22,36)(23,37)(29,38)(30,34)(31,39);(10,21)(15,29)(20,35)(22,23)(28,38)(30,31)(34,39)(36,37);(8,12,14,13)(9,17,19,18)(10,22,21,23)(11,25,26,27)(15,30,29,31)(16,24,32,33)(20,36,35,37)(28,34,38,39)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53415);