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×#62999 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
7114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 6 generators on 27 points:
(1,2,3)(4,5,8,7)(6,10,11,9)(12,13)(14,18)(15,19)(16,17)(20,26)(21,24)(22,27)(23,25);(4,6,8,11)(5,9,7,10);(4,7,8,5)(6,10,11,9)(13,17)(18,24)(19,25)(26,27);(12,14,16,21)(13,18,17,24)(15,22,23,20)(19,27,25,26);(12,15,16,23)(13,19,17,25)(14,20,21,22)(18,26,24,27);(12,13)(14,18)(15,19)(16,17)(20,26)(21,24)(22,27)(23,25)255 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55987);