Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C3×#61060 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#525 ◁ G | |
2686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 39 points:
(1,2,3)(4,5)(6,7)(8,9)(10,18)(11,15)(12,26)(13,16)(14,21)(17,27)(19,34)(20,30)(22,29)(23,37)(24,38)(25,32)(28,36)(31,35)(33,39);(8,10,13,22)(9,14,16,30)(11,23,25,19)(12,20,27,21)(15,31,32,28)(17,29,26,18)(24,36,39,35)(33,34,38,37);(5,7)(8,11,13,25)(9,15,16,32)(10,19,22,23)(12,24,27,39)(14,28,30,31)(17,33,26,38)(18,34,29,37)(20,35,21,36);(10,20)(12,27)(14,29)(17,26)(18,30)(19,35)(21,22)(23,36)(24,39)(28,37)(31,34)(33,38);(9,16)(14,30)(15,32)(17,26)(18,29)(28,31)(33,38)(34,37)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54048);