Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4×C8 · Schur multiplier (H2) = C25 |
| direct product | C2×C3×#1282 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C22⋊C4 |
| commutator | C22 | C2×C3×C4×C8 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| socle | C23×C3 | #95 |
| prime radical | C25×C3 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ C24×C3×C4 ◁ G | |
1130 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)(5,9,10,7)(6,8)(11,12,14,17,16,18,15,13);(3,5)(4,7)(6,10)(8,9)(11,13,15,18,16,17,14,12)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 911);