Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1355 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C2 |
| direct product | C3×#7266 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #329 |
| commutator | C42 | C3×C42 |
| Frattini, cosocle | C2×#97 | C22×C3 |
| prime radical | C23×C3 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#97 ◁ G | |
218 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 218 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,2,3)(4,5,6,7)(9,13)(10,17)(11,19)(14,22)(15,16)(18,20);(8,9,10,14,11,16,17,23,12,15,18,22,19,13,20,21)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 254);