Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C3×#60422 |
| 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 ◁ C2×C3×#117 ◁ G | |
2714 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)(10,15)(11,23)(12,17)(13,26)(14,18)(16,24)(19,33)(20,29)(21,30)(22,36)(25,31)(27,32)(28,35)(34,39)(37,38);(4,5,6,7)(8,10)(9,15)(11,19)(12,21)(13,22)(14,20)(16,28)(17,30)(18,29)(23,33)(24,35)(25,34)(26,36)(27,37)(31,39)(32,38);(8,11)(9,16)(10,19)(12,24)(13,27)(14,25)(15,28)(17,31)(18,32)(20,34)(21,35)(22,37)(23,26)(29,38)(30,39)(33,36);(10,20)(15,29)(19,34)(21,22)(28,38)(30,36)(33,39)(35,37);(8,12,14,13)(9,17,18,26)(10,21,20,22)(11,24,25,27)(15,30,29,36)(16,31,32,23)(19,35,34,37)(28,39,38,33)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53410);