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×#25529 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
1014 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)(7,9)(8,11)(12,13)(14,19)(15,26)(16,30)(17,21)(18,22)(20,29)(23,37)(24,35)(25,36)(27,42)(28,39)(31,34)(32,38)(33,41)(40,43);(4,6,7,11)(5,8,9,10)(12,14,18,25)(13,19,22,36)(15,27,28,41)(16,24,17,31)(20,37,38,43)(21,34,30,35)(23,32,40,29)(26,42,39,33);(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);(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, 18517);