Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C16 · Schur multiplier (H2) = C2×C4 |
| direct product | C3×#7145 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C42 | C42 |
| commutator | C4 | C3×C4×C16 |
| Frattini, cosocle | C2×C4×C8 | C22×C3 |
| socle | C22×C3 | #336 |
| prime radical | C23×C3 | C4⋊C8 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C3×C42 ◁ G | |
222 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 222 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,3)(4,5)(6,13,15,19)(7,11)(8,17)(9,12)(10,14,18,16)(20,21,22,25,23,26,28,31,24,27,29,32,30,33,34,35);(4,6,8,16,9,15,7,14)(5,10,11,13,12,18,17,19)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 133);