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×#23403 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
1942 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)(10,19)(11,16)(12,17)(13,26)(14,18)(15,21)(20,35)(22,29)(23,32)(24,34)(25,33)(27,30)(28,38)(31,36)(37,39);(8,10)(9,15)(11,20)(12,21)(13,27)(14,22)(16,31)(17,29)(18,30)(19,26)(23,36)(24,38)(25,37)(28,33)(32,39)(34,35);(5,7)(8,11,14,25)(9,16,18,33)(10,20,22,37)(12,24,13,23)(15,28,30,31)(17,34,26,32)(19,35,29,39)(21,38,27,36);(8,12,14,13)(9,17,18,26)(10,21,22,27)(11,23,25,24)(15,29,30,19)(16,32,33,34)(20,36,37,38)(28,35,31,39)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 16391);