Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C64 · Schur multiplier (H2) = C22 |
| direct product | C3×C22⋊C64 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C32 | C22 |
| commutator | C2 | C2×C3×C64 |
| Frattini, cosocle | C2×C32 | C22×C3 |
| socle | C22×C3 | C2×C32 |
| prime radical | C23×C3 | C32 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C32 ◁ G | |
98 conjugacy classes of subgroups.
| minimal degree | μ(G) = 71 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 2 generators on 71 points:
(1,2,3)(4,5)(6,7)(8,9,10,15,11,16,20,30,12,17,21,31,24,34,40,50,13,18,22,32,25,35,41,51,27,37,43,53,46,56,60,65,14,19,23,33,26,36,42,52,28,38,44,54,47,57,61,66,29,39,45,55,48,58,62,67,49,59,63,68,64,69,70,71);(5,7)480 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 500);