Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C2×C3×#2209 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×C8 ◁ G | |
702 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2)(39,40,41);(4,6)(7,8,12,16)(9,17,20,36)(10,21,25,35)(11,26,23,15)(13,28,30,18)(14,31,33,24)(19,37,27,32)(22,34,38,29);(3,4,5,6);(7,9,12,20)(8,13,16,30)(10,22,25,38)(11,27,23,19)(14,32,33,37)(15,34,26,29)(17,35,36,21)(18,31,28,24)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26565);