Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61629 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#61869 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C2×C3×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#533 ◁ G | |
1644 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 35 points:
(1,2,3)(4,5)(6,12)(7,10)(8,11)(9,14)(13,18)(15,17)(16,19)(20,21,25,29)(22,26,31,35)(23,28,24,33)(27,32,34,30);(4,6,8,14)(5,9,11,12)(7,15,16,13)(10,18,19,17)(20,22)(21,26)(23,32)(24,30)(25,31)(27,33)(28,34)(29,35);(20,22)(21,27)(23,30)(24,32)(25,31)(26,33)(28,35)(29,34);(4,7,8,16)(5,10,11,19)(6,13,14,15)(9,17,12,18)(20,23,25,24)(21,28,29,33)(22,30,31,32)(26,27,35,34);(5,11)(9,12)(10,19)(17,18)(20,23,25,24)(21,28,29,33)(22,30,31,32)(26,27,35,34)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54857);