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) = C27 |
| direct product | C3×#36172 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C3×#2416 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1910 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)(4,5,8,10)(6,7,9,11)(12,13)(14,19)(15,18)(16,17)(20,21,24,23)(22,26,27,25);(4,6)(5,7)(8,9)(10,11)(13,17)(14,18)(20,22,24,27)(21,25,23,26);(4,7)(5,9)(6,10)(8,11)(12,14,16,18)(13,15,17,19)(20,21,24,23)(22,25,27,26);(4,6)(5,7)(8,9)(10,11)(12,15,16,19)(13,18,17,14)(20,23,24,21)(22,26,27,25);(4,7)(5,9)(6,10)(8,11)(12,13)(14,15)(16,17)(18,19)210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 29160);