Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C26×C4 |
| direct product | C3×#36209 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C22×C3×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1802 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)(4,5,9,7)(6,10,11,8)(12,13,16,18)(14,15,17,19)(21,25)(26,27);(4,6)(5,8)(7,10)(9,11)(12,14)(13,15)(16,17)(18,19)(20,21,24,25)(22,26,23,27);(5,7)(6,11)(12,15)(13,17)(14,18)(16,19)(20,22,24,23)(21,26,25,27);(4,7,9,5)(6,10,11,8)(12,14)(13,15)(16,17)(18,19);(4,8,9,10)(5,6,7,11)(12,15)(13,17)(14,18)(16,19)(20,23,24,22)(21,27,25,26)210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 29197);