Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30437 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#30437 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #2668 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ G | |
906 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)(6,16)(7,21)(8,14)(9,13)(10,15)(11,19)(12,22)(17,28)(18,27)(20,26)(23,30)(24,29)(25,34)(31,33)(32,35);(4,6)(5,11)(7,17)(8,18)(9,19)(10,20)(12,28)(13,26)(14,16)(15,27)(21,31)(22,32)(23,34)(24,33)(25,29)(30,35);(4,7)(5,12)(6,17)(8,22)(9,24)(10,23)(11,25)(13,21)(14,30)(15,29)(16,31)(18,32)(19,33)(20,34)(26,35)(27,28);(6,18)(8,9)(11,26)(13,14)(16,27)(17,32)(19,20)(21,30)(22,24)(25,35)(28,31)(33,34)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 23425);