Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C3×#61078 |
| 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 | |
2694 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)(8,9,14,18)(10,19,22,29)(11,16,13,26)(12,17,25,34)(15,27,31,20)(21,36,38,33)(23,32,28,35)(24,39,37,30);(8,10)(9,15)(11,20)(12,24)(13,27)(14,22)(16,29)(17,33)(18,31)(19,26)(21,23)(25,37)(28,38)(30,35)(32,39)(34,36);(5,7)(8,11,14,13)(9,16,18,26)(10,20,22,27)(12,23,25,28)(15,29,31,19)(17,32,34,35)(21,37,38,24)(30,33,39,36);(8,12,14,25)(9,17,18,34)(10,21,22,38)(11,23,13,28)(15,30,31,39)(16,32,26,35)(19,36,29,33)(20,37,27,24);(9,18)(10,22)(16,26)(17,34)(20,27)(21,38)(24,37)(32,35)165 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54066);