Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23444 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#23446 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24×C3 |
| prime radical | C3×#3449 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
1994 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)(10,14)(11,15)(12,16)(13,17)(18,25)(19,26)(20,27)(21,29)(22,30)(23,31)(24,32)(28,34)(33,37)(35,38)(36,39);(8,10,13,20)(9,14,17,27)(11,18,22,34)(12,23,24,19)(15,28,30,25)(16,31,32,26)(21,35,36,33)(29,37,39,38);(5,7)(8,11)(9,15)(10,18)(12,21)(13,22)(14,25)(16,29)(17,30)(19,33)(20,34)(23,35)(24,36)(26,37)(27,28)(31,38)(32,39);(8,12,13,24)(9,16,17,32)(10,19,20,23)(11,21,22,36)(14,26,27,31)(15,29,30,39)(18,33,34,35)(25,37,28,38)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 16434);