Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C3×#61064 |
| 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×#525 ◁ G | |
2682 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)(10,20,21,34)(11,16,26,33)(12,18,13,17)(15,27,29,23)(22,36,37,31)(24,39,38,30)(25,35,28,32);(8,10,14,21)(9,15,19,29)(11,24,26,38)(12,23,13,27)(16,31,33,36)(17,20,18,34)(22,28,37,25)(30,35,39,32);(5,7)(10,21)(15,29)(20,34)(22,37)(23,27)(24,38)(30,39)(31,36);(8,11)(9,16)(10,22)(12,25)(13,28)(14,26)(15,30)(17,32)(18,35)(19,33)(20,36)(21,37)(23,38)(24,27)(29,39)(31,34);(8,12,14,13)(9,17,19,18)(10,23,21,27)(11,25,26,28)(15,20,29,34)(16,32,33,35)(22,38,37,24)(30,36,39,31)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54052);