Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#10873 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #121 |
| commutator | C2×C4 | C2×C3×C42 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| socle | C22×C3 | #389 |
| prime radical | C24×C3 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ G | |
714 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3)(4,5,6,7)(8,9,12,18,14,19,13,17)(10,20,23,31,24,30,22,15)(11,25,26,16,28,32,27,33)(21,29,36,34,37,39,35,38);(8,10,13,22,14,24,12,23)(9,15,17,30,19,31,18,20)(11,21,27,35,28,37,26,36)(16,29,25,38,33,39,32,34);(8,11,12,26,14,28,13,27)(9,16,17,25,19,33,18,32)(10,21,23,36,24,37,22,35)(15,29,30,38,31,39,20,34)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3861);