Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18252 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#18276 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
966 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)(4,5)(6,10)(7,9)(8,11)(12,15)(13,18)(14,16)(17,19)(25,31)(32,35);(5,9)(7,15)(10,16)(11,12)(14,19)(17,18)(20,21)(22,25)(23,26)(24,27)(28,32)(29,31)(30,33)(34,35);(4,6,8,13)(5,10,11,18)(7,14,15,19)(9,16,12,17)(20,22)(21,25)(23,28)(24,29)(26,32)(27,31)(30,34)(33,35);(5,11)(6,13)(9,12)(14,19)(20,23,24,30)(21,26,27,33)(22,28,29,34)(25,32,31,35)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 11264);