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) = C25×C4 |
| direct product | C2×C3×#1977 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | #121 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C23×C3×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C23×C3×C4 ◁ G | |
1566 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);(4,6)(7,8,12,16,13,15,11,9)(10,18,20,22,21,17,19,14);(3,4,5,6)(7,9,12,15,13,16,11,8)(10,17,20,22,21,18,19,14);(7,10,11,19,13,21,12,20)(8,14,16,18,15,22,9,17)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 14233);