Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24283 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#27676 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
704 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)(4,5,7,9)(6,8,10,11)(12,13,19,14)(15,23,25,27)(16,17,21,18)(20,26,22,24);(4,6)(5,8)(7,10)(9,11)(12,14,17,21,19,13,18,16)(15,20,24,23,25,22,26,27);(6,10)(8,11)(12,15)(13,20)(14,22)(16,23)(17,24)(18,26)(19,25)(21,27);(12,16,17,14,19,21,18,13)(15,23,24,22,25,27,26,20)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 20664);