Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×#8185 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C42 | C2×D4 |
| commutator | C4 | C2×C3×C4×C8 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| socle | C22×C3 | C4×C22⋊C4 |
| prime radical | C2×C3×#41 | C2×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C42 ◁ C2×C3×C4×C8 ◁ G | |
514 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)(12,13,16,20,17,21,15,18)(14,19,24,26,22,27,23,25);(4,5,6,8,7,9,10,11)(13,18,21,20)(14,22)(19,26,27,25)(23,24);(12,14,16,24,17,22,15,23)(13,19,20,26,21,27,18,25)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1173);