Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24468 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#25691 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
920 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)(4,5,7,8)(6,10,11,9)(12,13,18,22)(14,19,25,36)(15,26,28,39)(16,30,17,21)(20,32,38,29)(23,37,40,43)(24,35,31,34)(27,42,41,33);(5,8)(9,10)(12,14)(13,19)(15,27)(16,24)(17,31)(18,25)(20,37)(21,34)(22,36)(23,29)(26,42)(28,41)(30,35)(32,40)(33,39)(38,43);(12,15,16,29,18,28,17,32)(13,20,21,26,22,38,30,39)(14,23,24,41,25,40,31,27)(19,33,34,37,36,42,35,43);(4,6,7,11)(5,9,8,10)(15,28)(20,38)(23,40)(26,39)(27,41)(29,32)(33,42)(37,43)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 18679);