Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23912 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C3×#23952 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
1326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(4,5)(6,10)(7,9)(8,11)(12,13)(14,18)(15,20)(16,19)(17,21)(22,25)(23,27)(24,26);(4,6,7,11)(5,8,9,10)(12,14)(13,18)(15,22)(16,24)(17,23)(19,26)(20,25)(21,27);(13,19)(15,16)(18,25)(20,21)(22,24)(26,27);(5,9)(8,10)(12,15,17,16)(13,20,21,19)(14,22,23,24)(18,25,27,26)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 16940);