Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×#13456 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23×C3 |
| prime radical | C23×C3 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#405 ◁ G | |
318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3)(4,5,8,10)(6,11,7,9)(12,13,15,20)(14,24,26,18)(16,19,29,35)(17,22,30,23)(21,37,36,31)(25,33,28,38)(27,42,41,34)(32,40,39,43);(4,6,8,7)(5,9,10,11)(12,14,15,26)(13,18,20,24)(16,27,29,41)(17,25,30,28)(19,34,35,42)(21,32,36,39)(22,38,23,33)(31,40,37,43);(4,7,8,6)(5,9,10,11)(13,19)(14,25)(16,31)(17,30)(18,32)(20,35)(21,23)(22,36)(24,39)(26,28)(27,41)(29,37)(33,42)(34,38)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6444);