Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7316 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×#7316 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #1304 |
| commutator | C24 | C3×C42 |
| Frattini, cosocle | C23×D4 | C22×C3 |
| prime radical | C23×C3×D4 | C22 |
| LCS | G ▷ C24 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C23×C3 ◁ C23×C3×D4 ◁ G | |
620 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 620 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 48 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,3)(4,5,7,10)(6,12,13,18)(8,11,16,19)(9,15,17,14);(4,6)(5,9,11,12)(7,15,16,13)(8,14)(10,18)(17,19)84 conjugacy classes — view the full character table.
| Fitting subgroup of | D3×#7316, #72231786, #72231787 |
| maximal subgroup of | C3×#139750, C3×#144364, D3×#7316, … |
| maximal normal subgroup of | C3×#139750, C3×#144364, D3×#7316, … |
| maximal quotient of | C2×C3×#7316 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 304);