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×#19390 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| prime radical | C22×C3×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(5,8)(6,10)(9,11)(12,13,16,20)(14,17,21,25)(15,18,23,19)(22,24,27,26);(4,5,6,8,7,9,10,11)(12,14)(13,17)(15,22)(16,21)(18,24)(19,26)(20,25)(23,27);(13,18)(15,23)(17,24)(19,20)(22,27)(25,26);(14,21)(17,25)(22,27)(24,26)147 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 12378);