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×#19090 |
| 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 | |
888 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(5,10)(7,8)(11,18)(12,13)(14,16)(17,19)(20,21)(22,28)(23,31)(24,26)(25,30)(27,32)(29,35)(33,34);(4,5,9,13)(6,11,15,19)(7,10,8,12)(14,17,16,18)(23,32)(27,31)(28,35)(30,33);(4,6)(5,11)(7,14)(8,16)(9,15)(10,17)(12,18)(13,19)(20,22)(21,25)(23,33)(24,29)(26,34)(27,35)(28,31)(30,32);(4,7,9,8)(5,12,13,10)(6,14,15,16)(11,18,19,17)(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, 12078);