Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12096 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#12140 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C4⋊C4 | C23×C3 |
| prime radical | C3×#1598 | C2 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3 ◁ C2×C3×C4⋊C4 ◁ G | |
478 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 478 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 192 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(5,9)(6,7)(10,11)(12,13)(14,16)(15,18)(17,19);(4,5)(6,9)(7,11)(8,10)(14,19);(4,6,8,7)(5,9,10,11)(13,16)(14,19)(17,18)84 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 5128);