Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24986 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#29233 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #555 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C2×C3 | #2572 |
| prime radical | C3×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×C4 ◁ G | |
532 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)(4,5,9,14,10,15,8,13)(6,16,20,27,18,31,19,11)(7,21,23,30,24,12,25,29)(17,28,22,35,33,32,34,26);(4,6)(5,11,15,27)(7,22,24,34)(8,19)(9,20)(10,18)(12,28)(13,31,14,16)(17,23,33,25)(21,32)(26,30)(29,35);(4,7)(5,12)(6,17)(8,23)(9,25)(10,24)(11,26)(13,29)(14,30)(15,21)(16,32)(18,33)(19,22)(20,34)(27,35)(28,31);(5,13,15,14)(6,18)(11,16,27,31)(12,29,21,30)(17,33)(19,20)(22,34)(26,32,35,28)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 22221);