Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1357 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C22 |
| direct product | C3×#7268 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #331 |
| commutator | C42 | C3×C42 |
| Frattini, cosocle | C2×C4×C8 | C22×C3 |
| prime radical | C23×C3 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×C4×C8 ◁ G | |
238 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 118 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 — 2 generators on 23 points:
(1,2,3)(4,5,6,7)(9,13)(10,17)(11,19)(14,21)(15,16)(18,23);(8,9,10,14,11,13,17,20,12,15,18,21,19,16,23,22)138 conjugacy classes — view the full character table.
| Fitting subgroup of | D3×#7268, #72258806, #72258812 |
| maximal subgroup of | C3×#136361, C3×#136868, C3×#134947, … |
| maximal normal subgroup of | C3×#136361, C3×#136868, C3×#134947, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 256);