Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#25712 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C22×C3×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C4×D4 ◁ G | |
1060 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3)(5,7)(8,9,11,12)(10,14,15,13)(16,17)(18,24)(19,30)(20,27)(21,26)(22,25)(23,29)(28,31);(9,12)(13,14)(16,18,21,29)(17,23,26,24)(19,20,28,22)(25,31,27,30);(4,5,6,7)(16,19,22,18,21,28,20,29)(17,24,27,30,26,23,25,31);(8,10,11,15)(9,13,12,14)(16,20,21,22)(17,25,26,27)(18,28,29,19)(23,31,24,30)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 18700);