Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24468 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#25594 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
1050 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3)(4,5)(6,10)(7,8)(9,11)(12,13,17,21)(14,18,23,32)(15,19,26,35)(16,28,30,20)(22,38,39,31)(24,37,29,33)(25,34,40,43)(27,42,41,36);(5,8)(9,10)(12,14)(13,18)(15,25)(16,29)(17,23)(19,34)(20,37)(21,32)(22,27)(24,30)(26,40)(28,33)(31,36)(35,43)(38,42)(39,41);(12,15)(13,19)(14,22,23,39)(16,27)(17,26)(18,31,32,38)(20,36)(21,35)(24,25,29,40)(28,42)(30,41)(33,34,37,43);(4,6,7,11)(5,9,8,10)(15,26)(19,35)(22,39)(25,40)(27,41)(31,38)(34,43)(36,42)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 18582);