Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C3×#60432 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C4×#41 ◁ G | |
2646 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 39 points:
(1,2,3)(5,7)(8,9,13,17)(10,18,21,31)(11,15,22,29)(12,24,26,16)(14,25,28,20)(19,32,33,39)(23,36,35,30)(27,37,38,34);(4,5,6,7);(8,10,13,21)(9,14,17,28)(11,19,22,33)(12,25,26,20)(15,27,29,38)(16,31,24,18)(23,37,35,34)(30,39,36,32);(8,11)(9,15)(10,19)(12,23)(13,22)(14,27)(16,30)(17,29)(18,32)(20,34)(21,33)(24,36)(25,37)(26,35)(28,38)(31,39);(11,22)(15,29)(19,33)(23,35)(27,38)(30,36)(32,39)(34,37)183 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53420);