Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30434 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#30435 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #2668 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#2010 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ G | |
872 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5)(6,14)(7,18)(8,22)(9,13)(10,16)(11,20)(12,25)(15,33)(17,30)(19,28)(21,31)(23,26)(24,29)(27,34)(32,35);(4,6)(5,10)(7,19)(8,23)(9,17)(11,28)(12,30)(13,26)(14,22)(15,20)(16,25)(18,27)(21,35)(24,34)(29,32)(31,33);(10,26)(14,30)(27,33)(28,32);(4,7,9,21)(5,11,13,29)(6,15,17,34)(8,24,25,20)(10,27,26,33)(12,31,22,18)(14,32,30,28)(16,35,23,19)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 23423);