Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C22×C3×#356 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | #99 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C2×Q8 | C24×C3 |
| prime radical | C24×C3 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C24×C3 ◁ C23×C3×Q8 ◁ G | |
1254 conjugacy classes of subgroups; only the 382 normal subgroups are shown.
This group has 1254 conjugacy classes of subgroups — too many to draw. Only its 382 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2)(21,22,23);(3,4);(5,6,9,13,10,14,7,15)(8,12,16,19,18,20,17,11);(5,7,10,9)(6,11,14,19)(8,17,18,16)(12,13,20,15)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26352);