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×#63005 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
7024 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 6 generators on 27 points:
(1,2,3)(5,8)(9,11)(12,13,17,18)(14,16,20,25)(15,19,24,26)(21,23,22,27);(4,5)(6,9)(7,8)(10,11);(4,6,7,10)(5,9,8,11)(13,18)(16,25)(19,26)(23,27);(12,14,17,20)(13,16,18,25)(15,22,24,21)(19,27,26,23);(12,15,17,24)(13,19,18,26)(14,21,20,22)(16,23,25,27);(12,16)(13,20)(14,18)(15,23)(17,25)(19,22)(21,26)(24,27)255 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55993);