Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#13412 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23×C3 |
| prime radical | C2×C3×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#405 ◁ G | |
430 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3)(5,6)(8,9)(10,15)(11,16)(12,29)(13,30)(14,19)(17,26)(18,27)(20,38)(21,34)(22,39)(23,32)(24,36)(25,33)(28,37)(31,35);(4,5)(6,7)(8,10,14,24)(9,15,19,36)(11,25,28,22)(12,20,26,21)(13,31,27,23)(16,33,37,39)(17,34,29,38)(18,32,30,35);(4,6,7,5)(9,16)(10,20)(11,26)(12,28)(13,27)(15,32)(17,18)(19,37)(21,24)(22,31)(23,25)(29,30)(34,38)(35,36)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6400);