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×#9960 |
| 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 | |
650 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)(4,5,8,11)(6,9,7,10)(12,13)(14,17)(15,18)(16,19)(20,23)(21,24)(22,26)(25,27);(5,9)(6,7)(10,11)(12,14,16,21)(13,17,19,24)(15,20,22,27)(18,25,26,23);(5,10)(6,7)(9,11)(12,15)(13,18)(14,20)(16,22)(17,23)(19,26)(21,27)(24,25)111 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 2948);