Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C22 |
| direct product | C3×#9608 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #121 |
| commutator | C2×C4 | C22×C3×C8 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| socle | C22×C3 | #389 |
| prime radical | C23×C3 | C4×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ G | |
306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(4,5,8,6,7,9,11,10)(13,18)(14,21)(15,16)(19,26)(20,24)(22,23);(4,6,8,9,7,10,11,5)(12,13)(14,19)(15,24)(16,18)(17,20)(21,25)(22,27)(23,26);(12,14,15,22,17,23,16,21)(13,19,24,27,20,26,18,25)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 2596);