Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24398 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#24675 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#121 |
| 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 | |
764 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)(12,13,17,22,16,21,18,23)(14,24,27,39,26,19,33,35)(15,28,31,37,30,38,32,20)(25,41,29,34,42,36,43,40);(5,8)(9,10)(12,14)(13,19)(15,29)(16,26)(17,27)(18,33)(20,36)(21,24)(22,35)(23,39)(25,32)(28,40)(30,43)(31,42)(34,38)(37,41);(12,15,16,30)(13,20,21,37)(14,25,26,42)(17,32,18,31)(19,34,24,40)(22,38,23,28)(27,43,33,29)(35,41,39,36);(4,6,7,11)(5,9,8,10)(13,21)(19,24)(20,37)(22,23)(28,38)(34,40)(35,39)(36,41)129 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 17663);