Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#25610 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#1999 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G | |
1158 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)(8,9)(10,14)(11,12)(13,15)(16,17)(18,25)(19,23)(20,29)(21,24)(22,27)(26,30)(28,31);(4,5)(6,7)(9,12)(13,14)(16,17)(18,22)(19,29)(20,23)(21,24)(25,27)(26,31)(28,30);(5,7)(16,18,20,27,21,28,19,26)(17,22,23,25,24,30,29,31);(8,10,11,15)(9,13,12,14)(16,19,21,20)(17,23,24,29)(18,26,28,27)(22,25,30,31)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 18598);