Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11505 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C25 |
| direct product | C3×#12699 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#99 |
| commutator | C22×C4 | C22×C3×C4 |
| Frattini, cosocle | C22×Q8 | C23×C3 |
| prime radical | C23×C3×D4 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C24×C3 ◁ C23×C3×Q8 ◁ G | |
1146 conjugacy classes of subgroups; only the 158 normal subgroups are shown.
This group has 1146 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 96 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(4,5)(6,7)(8,9,11,15,12,16,20,23)(10,14,18,22,17,13,19,21);(9,13)(10,17)(14,16)(15,22)(18,19)(21,23);(5,7)120 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#139787, C3×#139785, C3×#139710, … |
| maximal normal subgroup of | C3×#139787, C3×#139785, C3×#139710, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 5687);