Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9946 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#9951 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #392 |
| commutator, Frattini, cosocle | C23×C4 | C23×C3 |
| prime radical | C2×C3×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
610 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(5,9)(6,10)(7,11)(12,13)(14,22)(15,18)(16,20)(17,21)(19,23)(24,26)(25,27);(4,5,8,6)(7,10,11,9)(13,18)(15,16)(19,22)(20,21)(23,25)(26,27);(4,6,8,5)(7,9,11,10)(12,14,17,24)(13,19,21,27)(15,23,16,25)(18,26,20,22)111 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 2939);