Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10260 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×#10651 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | #121 |
| commutator, Frattini, cosocle | C2×C42 | C23×C3 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G | |
350 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(4,5)(6,11)(7,9)(8,10)(12,13,14,17,15,18,19,16)(21,24)(25,26);(4,6,8,7)(5,9,10,11)(20,21,22,24)(23,25,27,26);(4,7,8,6)(5,9,10,11)(13,16)(14,19)(17,18)(21,25)(23,27)(24,26)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3639);