Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60367 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C210 |
| direct product | C3×#60370 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C2×C3×#585 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×#41 ◁ G | |
4746 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)(4,5)(6,7)(8,9,14,19)(10,20,23,33)(11,16,24,30)(12,18,13,17)(15,27,29,22)(21,34,35,39)(25,32,26,31)(28,37,38,36);(8,10)(9,15)(11,21)(12,22)(13,27)(14,23)(16,28)(17,20)(18,33)(19,29)(24,35)(25,36)(26,37)(30,38)(31,34)(32,39);(5,7)(8,11)(9,16)(10,21)(12,25)(13,26)(14,24)(15,28)(17,31)(18,32)(19,30)(20,34)(22,36)(23,35)(27,37)(29,38)(33,39);(11,24)(16,30)(21,35)(25,26)(28,38)(31,32)(34,39)(36,37);(8,12,14,13)(9,17,19,18)(10,22,23,27)(11,25,24,26)(15,20,29,33)(16,31,30,32)(21,36,35,37)(28,34,38,39)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53358);