Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#29840 |
| 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 | |
754 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)(4,5,6,7)(8,9,11,12)(10,13,14,15)(16,17)(18,22)(19,28)(20,23)(21,24)(25,29)(26,30)(27,31);(9,12)(13,15)(16,18,21,27)(17,22,24,31)(19,26,20,29)(23,25,28,30);(5,7)(10,14)(13,15)(16,17)(18,25)(19,23)(20,28)(21,24)(22,29)(26,31)(27,30);(8,10,11,14)(9,13,12,15)(16,19,21,20)(17,23,24,28)(18,26,27,29)(22,25,31,30)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 22828);