Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#30432 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G | |
886 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(4,5)(6,10)(7,9)(8,11)(12,13,17,22)(14,18,19,26)(15,24,25,23)(16,21,27,20);(4,6)(5,8)(7,11)(9,10)(12,14)(13,18)(15,21)(16,23)(17,19)(20,25)(22,26)(24,27);(12,15,17,25)(13,19,22,14)(16,26,27,18)(20,23,21,24);(5,9)(6,11)(12,14)(13,15)(16,24)(17,19)(18,21)(20,26)(22,25)(23,27)111 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 23420);