Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C3×#60450 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| socle | C22×C3 | C23×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C22×C3×#41 ◁ G | |
2722 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)(10,19)(11,16)(12,17)(13,27)(14,18)(15,22)(20,32)(21,29)(23,33)(24,30)(25,31)(26,37)(28,36)(34,39)(35,38);(8,10,13,22,14,23,12,21)(9,15,17,19,18,29,27,33)(11,20,26,36,24,34,25,35)(16,28,31,32,30,38,37,39);(5,7)(8,11)(9,16)(10,20)(12,25)(13,26)(14,24)(15,28)(17,31)(18,30)(19,32)(21,35)(22,36)(23,34)(27,37)(29,38)(33,39);(11,24)(16,30)(20,34)(25,26)(28,38)(31,37)(32,39)(35,36);(8,12,14,13)(9,17,18,27)(10,21,23,22)(11,25,24,26)(15,19,29,33)(16,31,30,37)(20,35,34,36)(28,32,38,39)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53438);