Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×C8×#101 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C8 | C22⋊C4 |
| commutator | C22 | C2×C3×C4×C8 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| socle | C22×C3 | C4×C22⋊C4 |
| prime radical | C23×C3 | C2×C42 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C8 ◁ C2×C3×C4×C8 ◁ G | |
454 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)(4,5,6,7,8,9,10,11);(12,13,15,19,16,18,24,26)(14,20,21,27,22,17,23,25);(12,14,16,22)(13,17,18,20)(15,23,24,21)(19,27,26,25)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1151);