Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24468 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#25619 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#3512 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
1052 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)(14,23)(15,27)(16,21)(17,31)(18,22)(19,25)(20,30)(24,41)(26,35)(28,34)(29,39)(32,36)(33,38)(37,42)(40,43);(5,8)(9,10)(12,14)(13,19)(15,28)(16,25)(17,26)(18,32)(20,37)(21,23)(22,35)(24,30)(27,40)(29,43)(31,36)(33,42)(34,39)(38,41);(12,15,17,30,16,29,18,33)(13,20,22,27,21,38,31,39)(14,24,26,43,25,42,32,28)(19,34,35,37,23,40,36,41);(4,6,7,11)(5,9,8,10)(13,21)(15,29)(19,23)(22,31)(24,42)(28,43)(30,33)(35,36)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 18607);