Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C2×C3×#2062 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | #120 |
| commutator, Frattini, cosocle | C24 | C24×C3 |
| prime radical | C25×C3 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C25×C3 ◁ G | |
1614 conjugacy classes of subgroups; only the 382 normal subgroups are shown.
This group has 1614 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) = 21 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2)(19,20,21);(3,4,7,9)(5,10,6,8)(11,12)(13,15)(14,16)(17,18);(3,5,7,6)(4,8,9,10)(11,13)(12,15)(14,17)(16,18);(3,6,7,5)(4,8,9,10)(15,18)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 16551);