Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8541 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C3×#8542 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #379 |
| commutator | C24 | C22×C3×C4 |
| Frattini, cosocle | C25 | C23×C3 |
| prime radical | C22×C3×#121 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C25×C3 ◁ G | |
1578 conjugacy classes of subgroups; only the 166 normal subgroups are shown.
This group has 1578 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 96 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(4,5)(6,10,11,8)(7,9)(12,13)(14,18,19,16)(15,17);(4,6,7,11)(5,8,9,10);(12,14)(13,16)(15,19)(17,18)120 conjugacy classes — view the full character table.
| Fitting subgroup of | D3×#8542, #72253488 |
| maximal subgroup of | C3×#245229, D3×#8542, #72253488 |
| maximal normal subgroup of | C3×#245229, D3×#8542, #72253488 |
| maximal quotient of | C3×#111516 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1530);