Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C213 |
| direct product | C3×#62986 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
7894 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 6 generators on 19 points:
(1,2,3)(4,5)(6,9)(7,8)(10,11)(12,13,16,15)(14,18,19,17);(12,14,16,19)(13,17,15,18);(5,8)(9,11);(6,10)(9,11)(12,13,16,15)(14,17,19,18);(4,6)(5,9)(7,10)(8,11)(12,15,16,13)(14,18,19,17);(4,5)(6,9)(7,8)(10,11)255 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55974);