Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9996 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#10001 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23×C3 |
| prime radical | C22×C3×D4 | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
568 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(4,5,8,11)(6,10,7,9)(12,13)(14,20)(15,23)(16,19)(17,22)(18,25)(21,27)(24,26);(5,9)(6,7)(10,11)(17,26)(20,27);(4,6,8,7)(5,10,11,9)(12,14)(13,17)(15,24)(16,21)(18,27)(19,26)(20,23)(22,25)111 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 2989);