Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19086 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#19088 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
884 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)(5,10)(6,13)(7,8)(12,15)(14,16)(17,19)(20,21)(22,28)(23,31)(24,26)(25,30)(27,32)(29,35)(33,34);(4,5)(6,11)(7,15)(8,10)(9,12)(13,19)(14,18)(16,17)(23,32)(27,31)(28,35)(30,33);(4,6,9,14)(5,11,12,18)(7,13,8,16)(10,17,15,19)(20,22)(21,25)(23,33)(24,29)(26,34)(27,35)(28,31)(30,32);(5,12)(10,15)(11,18)(17,19)(21,26)(25,34)(27,31)(28,35)93 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 12076);