Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C8 · Schur multiplier (H2) = C2×C4 |
| direct product | C3×#7021 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | #95 |
| commutator | C2×C4 | C3×C4×C8 |
| Frattini, cosocle | C22×C42 | C22×C3 |
| prime radical | C24×C3 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G | |
380 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3)(4,5)(6,8)(7,10)(9,11)(12,13)(14,18,19,16)(15,17)(20,21,23,26,25,27,24,22);(5,8,10,9)(12,14)(13,16)(15,19)(17,18)(20,22,24,27,25,26,23,21)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 9);