Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#19415 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| prime radical | C2×C3×#41 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
640 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(4,5,8,10)(6,11,9,7)(12,13,17,21)(14,18,23,27)(15,20,16,25)(19,24,26,22);(4,6,7,5,8,9,11,10)(12,14)(13,18)(15,24)(16,22)(17,23)(19,25)(20,26)(21,27);(12,14)(13,19)(15,22)(16,24)(17,23)(18,25)(20,27)(21,26);(12,15,17,16)(13,20,21,25)(14,22,23,24)(18,19,27,26)147 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 12403);