Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C8 · Schur multiplier (H2) = C2×C4 |
| direct product | C3×#7206 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #99 |
| commutator | C2×C4 | C3×C4×C8 |
| Frattini, cosocle | C2×C4×Q8 | C22×C3 |
| socle | C22×C3 | #342 |
| prime radical | C23×C3 | #95 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C22×C3×C4 ◁ C2×C3×C4×Q8 ◁ G | |
284 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 284 conjugacy classes of subgroups — too many to draw. Only its 102 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)(4,5,7,10,9,11,8,6)(12,13,15,19,16,20,24,27)(14,18,21,26,22,17,23,25);(4,6,8,11,9,10,7,5)(13,17,20,18)(19,26,27,25)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 194);