Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7262 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C22 |
| direct product | C3×#7264 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #340 |
| commutator | C2×C8 | C3×C42 |
| Frattini, cosocle | C23×C8 | C22×C3 |
| prime radical | C24×C3 | SD16 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ C23×C3×C8 ◁ G | |
374 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 374 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 96 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3)(4,5)(6,8)(7,10)(9,11)(12,13,15,20,17,22,16,21)(14,23,25,19,26,18,24,27);(5,8,10,9)(13,18)(14,24)(15,16)(19,20)(21,27)(22,23)(25,26)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 252);