Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#13329 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C2×D8 |
| commutator, Frattini, cosocle | C22×C8 | C23×C3 |
| prime radical | C22×C3×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×C8 ◁ G | |
578 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)(6,7)(12,13,17,22)(14,23,26,41)(15,19,30,37)(16,31,29,21)(18,33,35,24)(20,38,36,28)(25,42,32,39)(27,40,43,34);(5,7)(8,9)(10,11)(12,14,17,26)(13,18,22,35)(15,27,30,43)(16,32,29,25)(19,23,37,41)(20,39,36,42)(21,40,31,34)(24,38,33,28);(9,11)(13,19)(15,28)(16,29)(18,23)(20,21)(22,37)(24,27)(25,32)(30,38)(31,36)(33,43)(34,42)(35,41)(39,40)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6317);