Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C2×C3×#2056 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C3 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ C22×C3×C42 ◁ G | |
1486 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2)(27,28,29);(3,4)(5,6)(11,12,16,20)(13,18,22,26)(14,23,21,17)(15,25,24,19);(7,8)(9,10)(11,13,16,22)(12,17,20,23)(14,24,21,15)(18,19,26,25);(4,6)(8,10)(11,14,15,13,16,21,24,22)(12,18,19,23,20,26,25,17)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 16526);