Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×C3×#1907 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | #121 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C22×C3×AGL(1,ℤ/8) | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C23×C3×C4 ◁ G | |
1826 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2)(23,24,25);(3,4)(5,6)(7,8,12,14,11,10,16,19)(9,15,17,22,18,13,20,21);(7,9,16,17,11,18,12,20)(8,13,14,22,10,15,19,21);(4,6)(7,10)(8,11)(9,15)(12,14)(13,18)(16,19)(17,21)(20,22)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13989);