Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32696 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#32696 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #2749 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ G | |
552 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)(4,5,8,13,9,12,23,29)(6,14,17,30,16,10,22,25)(7,18,21,34,20,11,26,28)(15,31,19,33,32,24,35,27);(4,6)(5,10)(7,19)(8,22)(9,16)(11,27)(12,14)(13,30)(15,21)(17,23)(18,33)(20,35)(24,28)(25,29)(26,32)(31,34);(4,7)(5,11,12,18)(6,15,16,32)(8,21)(9,20)(10,24)(13,28,29,34)(14,31)(17,19,22,35)(23,26)(25,27)(30,33);(5,12)(8,23)(11,18)(17,22)(19,35)(21,26)(25,30)(27,33)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 25684);