Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#30486 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#3008 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G | |
1024 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3)(4,5)(6,7)(8,9,12,18,14,20,13,19)(10,21,23,33,25,32,24,15)(11,16,27,34,28,35,29,17)(22,31,26,36,37,39,38,30);(8,10,14,25)(9,15,20,33)(11,26,28,38)(12,24,13,23)(16,31,35,39)(17,36,34,30)(18,32,19,21)(22,29,37,27);(5,7)(8,11)(9,16)(10,22)(12,27)(13,29)(14,28)(15,30)(17,19)(18,34)(20,35)(21,31)(23,26)(24,38)(25,37)(32,39)(33,36);(5,7)(8,11)(9,17)(10,22)(12,29)(13,27)(14,28)(15,31)(16,19)(18,35)(20,34)(21,30)(23,38)(24,26)(25,37)(32,36)(33,39)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 23474);