Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18562 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#18604 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| prime radical | C3×#3214 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
900 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)(4,5)(6,13)(7,11)(8,17)(9,12)(10,14)(15,19)(16,18)(21,25)(22,28)(23,30)(27,31)(29,32)(33,35);(4,6,9,16)(5,10,12,19)(7,14,8,15)(11,18,17,13)(20,21,24,27)(22,26,29,34)(23,31,30,25)(28,33,32,35);(4,7,9,8)(5,11,12,17)(6,14,16,15)(10,18,19,13)(20,22,24,29)(21,26,27,34)(23,32,30,28)(25,33,31,35);(4,7,9,8)(5,11,12,17)(6,15,16,14)(10,13,19,18)(21,27)(25,31)(26,34)(33,35)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 11592);