Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C16 · Schur multiplier (H2) = C22 |
| direct product | C3×#7163 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4×C8 | D4 |
| commutator | C4 | C3×C4×C16 |
| Frattini, cosocle | C2×C4×C8 | C22×C3 |
| socle | C22×C3 | #336 |
| prime radical | C23×C3 | C4⋊C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4×C8 ◁ C2×C3×C4×C8 ◁ G | |
238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3)(4,5)(6,8)(7,10)(9,11)(12,13,15,20,17,14,19,23,18,22,16,21,26,25,27,24);(5,8,10,9)(12,14,16,24,17,22,27,20,18,25,15,23,26,13,19,21)336 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 151);