Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24 |
| direct product | C3×#17958 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C8 | C2×D4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C42 | C24×C3 |
| socle | C22×C3 | C2×C4×D4 |
| prime radical | C23×C3 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C8 ◁ C2×C3×C4×C8 ◁ G | |
614 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5,8,11)(6,12,14,19)(7,15,17,10)(9,16,18,13)(20,21,25,28,26,22,24,30)(23,27,32,34,33,29,31,35);(4,6,8,14)(5,9,11,18)(7,16,17,13)(10,19,15,12)(20,22,26,21)(23,29,33,27)(24,28,25,30)(31,34,32,35);(20,23,24,31,26,33,25,32)(21,27,30,35,22,29,28,34);(21,22)(27,29)(28,30)(34,35)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 10946);