Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4×C16 · Schur multiplier (H2) = C23 |
| direct product | C2×C3×#1397 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C16 | C22 |
| commutator | C2 | C2×C3×C4×C16 |
| Frattini, cosocle | C2×C16 | C23×C3 |
| socle, prime radical | C23×C3 | C2×C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C16 ◁ G | |
254 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2)(39,40,41);(7,8,9,14,10,15,17,26,11,16,18,27,20,28,29,36,12,13,19,23,21,24,30,33,22,25,31,34,32,35,37,38);(3,4,5,6)(8,13)(14,23)(15,24)(16,25)(26,33)(27,34)(28,35)(36,38)480 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6535);