Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24378 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#29219 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C3×C43 ◁ G | |
778 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(4,5,8,11)(6,10,7,9)(12,13,15,17)(14,16,18,19)(21,24)(22,25)(26,27);(4,6,8,7)(5,9,11,10)(12,14,15,18)(13,16,17,19)(20,21,23,24)(22,25,26,27);(4,6,8,7)(5,10,11,9)(14,18)(16,19)(20,22,24,27,23,26,21,25);(4,7,8,6)(5,10,11,9)(20,21,23,24)(22,25,26,27)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 22207);