Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23400 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#23401 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
1954 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)(10,18)(11,15)(12,24)(13,17)(14,20)(16,27)(19,35)(21,33)(22,31)(23,34)(25,29)(26,32)(28,38)(30,37)(36,39);(8,10,13,21)(9,14,17,29)(11,19,22,36)(12,25,27,20)(15,30,31,28)(16,33,24,18)(23,38,26,37)(32,39,34,35);(5,7)(8,11)(9,15)(10,19)(12,26)(13,22)(14,28)(16,34)(17,31)(18,35)(20,38)(21,36)(23,27)(24,32)(25,37)(29,30)(33,39);(11,22)(15,31)(19,36)(23,26)(28,30)(32,34)(35,39)(37,38)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 16389);