Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C3×#9869 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C2×D4 |
| commutator | C22×C4 | C22×C3×C4 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| socle, prime radical | C23×C3 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ C3×C43 ◁ G | |
512 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(4,5)(6,10)(7,8)(9,11)(13,17,18,19)(20,21,23,26,24,27,22,25);(5,8)(9,10)(12,13,14,17,16,18,15,19)(21,25)(22,23)(26,27);(4,6,7,11)(5,9,8,10)(12,14,16,15)(13,17,18,19)(20,22,24,23)(21,25,27,26)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 2857);