Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C43 · Schur multiplier (H2) = C22×C42 |
| direct product | C3×#10706 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C8 | D4 |
| commutator | C4 | C3×C43 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| socle | C23×C3 | C4×D4 |
| prime radical | C24×C3 | #41 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C8 ◁ C23×C3×C8 ◁ G | |
926 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)(6,7)(8,9,10,11)(12,13)(14,19)(15,18)(16,20)(17,21)(22,25)(23,26)(24,27);(5,7)(13,18,21,20)(19,25,27,26);(12,14,16,23,17,24,15,22)(13,19,20,26,21,27,18,25)336 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3694);