Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C27 |
| direct product | C2×C3×#2001 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C25×C3 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ C24×C3×C4 ◁ G | |
2182 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2)(27,28,29);(3,4)(5,6)(11,12,15,18)(13,19,21,26)(14,22,24,17)(16,23,25,20);(7,8)(9,10)(11,13,15,21)(12,16,18,25)(14,23,24,20)(17,26,22,19);(4,6)(8,10)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 16373);