Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42×C8 · Schur multiplier (H2) = C2×C42 |
| direct product | C3×C8×#97 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4×C8 | C22 |
| commutator | C2 | C3×C42×C8 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| socle, prime radical | C23×C3 | C2×C42 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4×C8 ◁ G | |
606 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(4,5,6,7,8,9,10,11);(16,17,18,21,19,20,22,23);(12,13,14,15)(17,20)(21,23)480 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 828);