Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7076 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C3×#7090 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #327 |
| commutator | C22×C8 | C2×C3×C4 |
| Frattini, cosocle | C2×C4⋊C8 | C22×C3 |
| prime radical | C23×C3 | #102 |
| LCS | G ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×C4⋊C8 ◁ G | |
212 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 192 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3)(4,5,6,9,7,8,10,11)(13,17)(14,20)(15,22)(18,27)(19,26)(21,24)(23,25);(5,8)(12,13,16,19)(14,21,23,27)(15,24,22,18)(17,25,26,20)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 78);