Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25 |
| direct product | C3×#33410 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D8 |
| commutator | C8 | C23×C3×C4 |
| Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C2×C3×#41 | D4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×#41 ◁ C2×C3×#131 ◁ G | |
658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3)(4,5,6,7)(9,14)(11,15)(12,22)(16,24)(17,32)(18,27)(20,28)(21,34)(23,29)(25,26)(30,35)(31,39)(33,38)(36,37);(8,9,15,26,12,17,23,27,13,18,29,32,22,25,11,14)(10,16,28,37,21,31,33,35,19,30,38,39,34,36,20,24);(8,10)(9,16)(11,20)(12,21)(13,19)(14,24)(15,28)(17,31)(18,30)(22,34)(23,33)(25,36)(26,37)(27,35)(29,38)(32,39);(10,19)(16,30)(20,33)(21,34)(24,35)(28,38)(31,36)(37,39)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26398);