Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25446 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#25446 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #2428 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ G | |
876 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5,9,13,10,14,8,25)(6,15,19,16,20,29,18,11)(7,21,23,27,24,34,22,12)(17,30,32,35,33,26,31,28);(4,6)(5,11,14,16)(7,17)(8,18)(9,19)(10,20)(12,26,27,30)(13,15,25,29)(21,28,34,35)(22,31)(23,32)(24,33);(4,5)(6,16)(7,12)(8,13)(9,25)(10,14)(11,20)(15,19)(17,30)(18,29)(21,22)(23,34)(24,27)(26,33)(28,32)(31,35);(4,7,8,22,10,24,9,23)(5,12,13,21,14,27,25,34)(6,17,18,31,20,33,19,32)(11,26,15,28,16,30,29,35)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 18434);