Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13323 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#13323 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #1999 |
| commutator, Frattini, cosocle | C22×C8 | C23×C3 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G | |
534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(5,11)(6,16)(7,20)(8,9)(12,29)(13,27)(14,21)(15,28)(17,30)(18,22)(19,25)(23,24)(26,31)(32,35);(4,5)(6,17)(7,21)(8,13)(9,14)(10,15)(11,20)(12,16)(18,26)(19,31)(22,30)(23,28)(24,27)(25,32)(29,33)(34,35);(4,6)(5,12)(7,22)(8,25)(9,16)(10,19)(11,26)(13,17)(14,31)(15,32)(18,24)(20,34)(21,35)(23,33)(27,29)(28,30)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6311);