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×#62990 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
7348 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 6 generators on 27 points:
(1,2,3)(5,8)(9,11)(12,13,17,18)(14,19,24,27)(15,20,25,23)(16,21,26,22);(4,5)(6,9)(7,8)(10,11);(4,6,7,10)(5,9,8,11)(13,18)(16,26)(19,27)(20,23);(12,14)(13,19)(15,21)(16,23)(17,24)(18,27)(20,26)(22,25);(12,15)(13,20)(14,22)(16,19)(17,25)(18,23)(21,24)(26,27);(12,16)(13,21)(14,23)(15,19)(17,26)(18,22)(20,24)(25,27)255 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55978);