Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(7,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3×C4 · Schur multiplier (H2) = C214 |
| direct product | C3×#63076 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C26 |
| commutator | C2 | C25×C3×C4 |
| Frattini, cosocle | C4 | C26×C3 |
| socle | C2×C3 | C25×C4 |
| prime radical | C2×C3×#585 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ G | |
10796 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 6 generators on 35 points:
(1,2,3)(4,5,8,13,9,12,19,25)(6,10,16,22,15,21,28,32)(7,11,18,24,17,23,29,33)(14,20,27,31,26,30,34,35);(4,6)(5,10)(7,14)(8,16)(9,15)(11,20)(12,21)(13,22)(17,26)(18,27)(19,28)(23,30)(24,31)(25,32)(29,34)(33,35);(4,7)(5,11)(6,14)(8,18)(9,17)(10,20)(12,23)(13,24)(15,26)(16,27)(19,29)(21,30)(22,31)(25,33)(28,34)(32,35);(7,17)(11,23)(14,26)(18,29)(20,30)(24,33)(27,34)(31,35);(6,15)(10,21)(14,26)(16,28)(20,30)(22,32)(27,34)(31,35);(5,12)(10,21)(11,23)(13,25)(20,30)(22,32)(24,33)(31,35)390 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56064);