Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×#8161 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C2×C22⋊C4 |
| commutator | C23 | C2×C3×C42 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| socle | C22×C3 | C4×C22⋊C4 |
| prime radical | C22×C3×D4 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C23×C3×C8 ◁ G | |
568 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(4,5)(6,10)(7,9)(8,11)(12,13,15,14,16,17,19,18);(4,6)(5,8,9,10)(7,11);(12,14,15,17,16,18,19,13)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1149);