Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11651 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C2×C4 |
| direct product | C3×#13279 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#102 |
| commutator | C2×C8 | C22×C3×C4 |
| Frattini, cosocle | C22×C8 | C23×C3 |
| prime radical | C23×C3 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×C8 ◁ G | |
366 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)(6,10)(7,8)(9,11)(12,13,18,22,25,19,16,26,17,21,27,24,15,14,23,20);(5,8)(9,10)(12,14,15,13,17,19,25,21)(16,26,18,24,23,20,27,22);(4,6,7,11)(5,9,8,10)(12,15,17,25)(13,19,21,14)(16,18,23,27)(20,22,26,24)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6267);