Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#33035 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C22×C3×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C3×#376 ◁ G | |
842 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)(5,6)(8,9)(10,13)(11,12)(14,15)(16,17)(18,27)(19,29)(20,24)(21,26)(22,23)(25,28)(30,31);(4,5,7,6)(9,12)(13,15)(16,18,21,22)(17,23,26,27)(19,30,20,28)(24,31,29,25);(4,6,7,5)(8,10,11,14)(9,13,12,15)(16,19,21,20)(17,24,26,29)(18,28,22,30)(23,25,27,31);(16,20,21,19)(17,25,26,31)(18,28,22,30)(23,24,27,29)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26023);