Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61643 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#61874 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C3×#3554 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#533 ◁ G | |
1654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 35 points:
(1,2,3)(4,5)(6,12)(7,10)(8,11)(9,14)(13,18)(15,17)(16,19)(20,21)(22,25)(23,31)(24,29)(26,28)(27,32)(30,34)(33,35);(4,6,8,14)(5,9,11,12)(7,15,16,13)(10,18,19,17)(20,22,25,21)(23,30,33,27)(24,29,26,28)(31,32,35,34);(20,23,25,33)(21,27,22,30)(24,32,26,34)(28,31,29,35);(4,7,8,16)(5,10,11,19)(6,13,14,15)(9,17,12,18)(20,24,25,26)(21,28,22,29)(23,32,33,34)(27,31,30,35);(5,11)(9,12)(10,19)(17,18)(21,22)(27,30)(28,29)(31,35)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54862);