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×#60433 |
| 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 | |
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)(10,18)(11,16)(12,14)(13,25)(15,21)(17,24)(19,32)(20,31)(22,28)(23,36)(26,27)(29,34)(30,35)(33,39)(37,38);(4,5,6,7)(9,14)(15,27)(16,28)(17,25)(18,31)(29,38)(30,36)(32,39);(8,10,12,20)(9,15,14,27)(11,19,22,33)(13,26,24,21)(16,29,28,38)(17,31,25,18)(23,37,35,34)(30,39,36,32);(8,11)(9,16)(10,19)(12,22)(13,23)(14,28)(15,29)(17,30)(18,32)(20,33)(21,34)(24,35)(25,36)(26,37)(27,38)(31,39);(11,22)(16,28)(19,33)(23,35)(29,38)(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, 53421);