Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25 |
| direct product | C3×#20774 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C24 |
| commutator | C23 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C23×C3 | C23×C4 |
| prime radical | C24×C3 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ G | |
818 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)(6,11)(7,9)(8,10)(12,13)(14,16)(15,17)(18,19)(20,21,23,26,24,27,30,33)(22,25,29,32,28,31,34,35);(4,6,8,7)(5,9,10,11)(12,14,15,18)(13,16,17,19)(20,22)(21,25)(23,29)(24,28)(26,32)(27,31)(30,34)(33,35);(4,7,8,6)(5,9,10,11)(13,17)(16,19);(22,28)(25,31)(29,34)(32,35)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13762);