Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4×C8 · Schur multiplier (H2) = C25×C4 |
| direct product | C22×C3×#322 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C42 | C22 |
| commutator | C2 | C22×C3×C4×C8 |
| Frattini, cosocle | C42 | C24×C3 |
| socle, prime radical | C24×C3 | C42 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C42 ◁ G | |
1182 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2)(21,22,23);(3,4);(5,6,7,10,8,9,11,12);(6,9)(10,12)(13,14,15,17,16,18,19,20)480 conjugacy classes — view the full character table.
| Fitting subgroup of | C22×D3×#322 |
| maximal subgroup of | C22×D3×#322 |
| maximal normal subgroup of | C22×D3×#322 |
| maximal quotient of | C3×#479908 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 10299);