Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9369 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C3×#9373 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #386 |
| commutator | C42 | C22×C3×C4 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| prime radical | C3×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(4,5,8,12,9,11,7,17)(6,13,16,18,15,19,14,10)(20,21,25,27,24,23,31,32)(22,28,29,35,30,26,33,34);(4,6,9,15)(5,10)(7,14,8,16)(11,18)(12,13)(17,19)(20,22,25,33,24,30,31,29)(21,26,32,34,23,28,27,35);(5,11)(7,8)(10,18)(14,16)(20,23)(21,24)(22,28)(25,27)(26,30)(29,34)(31,32)(33,35)111 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 2361);