Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24398 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C3×#24403 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#121 |
| 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 | |
1332 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)(8,9,12,17,14,19,13,18)(10,15,21,30,23,32,22,31)(11,24,28,16,27,35,26,34)(20,36,25,29,38,33,37,39);(5,7)(8,10)(9,15)(11,25)(12,21)(13,22)(14,23)(16,33)(17,30)(18,31)(19,32)(20,26)(24,29)(27,37)(28,38)(34,36)(35,39);(8,11)(9,16)(10,20)(12,26)(13,28)(14,27)(15,29)(17,24)(18,35)(19,34)(21,37)(22,25)(23,38)(30,36)(31,33)(32,39);(5,7)(8,12,14,13)(9,17,19,18)(10,21,23,22)(11,26,27,28)(15,30,32,31)(16,24,34,35)(20,37,38,25)(29,36,39,33)129 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 17391);