Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#15097 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C3 | C24 |
| commutator, Frattini, cosocle | C24 | C24×C3 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
1374 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(5,6)(8,9,10,11)(14,18)(17,19)(20,21,23,25)(22,24,26,27);(4,5,7,6)(13,16)(17,19)(20,21)(22,24)(23,25)(26,27);(8,9)(10,11)(12,13,15,16)(14,17,18,19)(20,21)(22,24)(23,25)(26,27);(4,6,7,5)(12,14)(13,17)(15,18)(16,19)(20,22,23,26)(21,24,25,27)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 8085);