Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#27070 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C22×C3×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C3×C4×C4⋊C4 ◁ G | |
740 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)(4,5,8,9)(6,10,7,11)(12,13,15,16)(14,18,19,17)(20,21,24,27)(22,25,30,34)(23,26,31,35)(28,32,29,33);(5,9)(10,11)(13,16)(17,18)(20,22)(21,25)(23,28)(24,30)(26,32)(27,34)(29,31)(33,35);(4,6,8,7)(5,10,9,11)(12,14,15,19)(13,17,16,18)(22,28)(23,31)(25,32)(26,35)(29,30)(33,34);(4,7,8,6)(5,10,9,11)(13,16)(17,18)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 20058);