Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C22×C4 |
| direct product | C2×C3×#1285 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | #95 |
| commutator | C2×C4 | C2×C3×C42 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| prime radical | C24×C3 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G | |
746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2)(19,20,21);(3,4,6,9,7,10,5,8)(11,12,13,16,14,15,17,18);(4,8,10,9)(12,15)(13,17)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 935);