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×#18593 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| prime radical | C3×#3511 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
916 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)(20,21,25,29)(22,30,33,34)(23,26,24,28)(27,32,35,31);(4,6,9,16)(5,10,12,19)(7,14,8,15)(11,18,17,13)(21,26)(23,24)(27,34)(28,29)(30,35)(31,32);(4,7,9,8)(5,11,12,17)(6,14,16,15)(10,18,19,13)(20,22,25,33)(21,27,29,35)(23,31,24,32)(26,34,28,30);(4,7,9,8)(5,11,12,17)(6,15,16,14)(10,13,19,18)(20,23,25,24)(21,26,29,28)(22,31,33,32)(27,34,35,30)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 11581);