Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11541 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24 |
| direct product | C3×#11557 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#99 |
| commutator | C24 | C22×C3×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C23×C3 |
| prime radical | C3×#2408 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×C4×D4 ◁ G | |
750 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 750 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(4,5,7,11,8,9,6,10)(12,13)(14,18,19,16)(15,17);(5,9)(6,7)(13,16)(17,18);(4,6,8,7)(5,10,9,11)111 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 4545);