Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C24 |
| direct product | C3×#7250 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | #99 |
| commutator | C22×C4 | C3×C42 |
| Frattini, cosocle | C22×C22⋊C4 | C22×C3 |
| prime radical | C25×C3 | D4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C24×C3 ◁ C22×C3×C22⋊C4 ◁ G | |
632 conjugacy classes of subgroups.
| 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)(8,9)(10,12)(11,14)(13,15)(17,20)(21,22);(9,12,14,13)(16,17)(18,22,23,20)(19,21)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 238);