Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C3×#37679 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
3064 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3)(9,11)(13,14)(15,22)(19,21)(23,27);(5,7)(12,13,17,14)(15,18,22,20)(16,21,24,19)(23,26,27,25);(4,5)(6,7)(8,9,10,11)(12,14,17,13)(15,20,22,18)(16,21,24,19)(23,26,27,25);(12,15)(13,18)(14,20)(16,23)(17,22)(19,25)(21,26)(24,27);(4,5)(6,7)(12,16,17,24)(13,19,14,21)(15,23,22,27)(18,25,20,26)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 30667);