Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12019 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C3×#12019 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#106 |
| commutator | C2×C8 | C22×C3×C4 |
| Frattini, cosocle | C4×C8 | C23×C3 |
| prime radical | C23×C3 | #106 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×C8 ◁ G | |
226 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)(4,5,6,9,7,8,10,11)(13,17,18,23)(14,20)(15,22)(19,26,25,24)(21,27);(12,13,14,17,15,19,20,24,16,18,21,23,22,25,27,26);(5,8)(9,11)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 5007);