Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33758 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#33784 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C3×#466 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×C8 ◁ G | |
584 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3)(4,5)(6,10)(7,9)(8,11)(12,13)(14,22)(15,18)(16,28)(17,31)(19,24)(20,30)(21,27)(23,38)(25,42)(26,39)(29,37)(32,36)(33,40)(34,41)(35,43);(4,6,7,11)(5,8,9,10)(13,18)(19,33)(20,34)(21,31)(22,38)(28,39)(35,37)(36,42);(6,11)(8,10)(12,14,15,23)(13,19,18,33)(16,29,26,43)(17,32,27,25)(20,36,34,42)(21,37,31,35)(22,39,38,28)(24,41,40,30);(5,9)(8,10)(13,18)(19,33)(20,34)(21,31)(22,38)(28,39)(35,37)(36,42)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26772);