Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C32 · Schur multiplier (H2) = C2 |
| direct product | C3×#7513 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C16 | C22⋊C4 |
| commutator | C22 | C2×C3×C32 |
| Frattini, cosocle | C2×C32 | C22×C3 |
| socle | C2×C3 | C22⋊C32 |
| prime radical | C23×C3 | C32 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C16 ◁ C2×C3×C32 ◁ G | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 67 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 2 generators on 67 points:
(1,2,3)(4,5,6,12,7,13,17,30,8,14,18,31,20,33,39,52,9,15,19,32,21,34,40,53,23,35,41,54,42,55,59,65,10,11,16,26,22,27,36,46,24,28,37,47,43,49,56,61,25,29,38,48,44,50,57,62,45,51,58,63,60,64,66,67);(5,11)(6,16)(13,27)(14,28)(15,29)(17,36)(18,37)(19,38)(33,49)(34,50)(35,51)(39,56)(40,57)(41,58)(55,64)(59,66)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 501);