Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#13387 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23×C3 |
| prime radical | C3×#3058 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G | |
428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3)(5,6)(8,9)(10,14)(11,15)(12,27)(13,18)(16,24)(17,25)(19,29)(20,37)(21,30)(22,31)(23,32)(26,34)(28,36)(33,39)(35,38);(4,5)(6,7)(8,10)(9,14)(11,23)(12,19)(13,22)(15,32)(16,35)(17,30)(18,31)(20,26)(21,25)(24,38)(27,29)(28,33)(34,37)(36,39);(4,6,7,5)(9,15)(10,19)(11,24)(12,25)(14,28)(16,17)(18,34)(20,23)(21,22)(26,39)(27,33)(29,37)(30,32)(31,35)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6375);