Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C2×C3×#3044 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G | |
2666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 41 points:
(1,2)(39,40,41);(4,6)(7,8,12,16)(9,13,18,27)(10,20,22,30)(11,23,25,15)(14,24,29,21)(17,31,32,38)(19,35,34,28)(26,36,37,33);(3,4,5,6)(7,9)(8,13)(10,17)(11,19)(12,18)(14,26)(15,28)(16,27)(20,31)(21,33)(22,32)(23,35)(24,36)(25,34)(29,37)(30,38);(7,10,12,22)(8,14,16,29)(9,17,18,32)(11,24,25,21)(13,26,27,37)(15,30,23,20)(19,36,34,33)(28,38,35,31);(9,18)(13,27)(17,32)(19,34)(26,37)(28,35)(31,38)(33,36)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53407);