Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18316 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#18317 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×#534 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
1402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(4,5,7,9)(6,8,11,10)(12,13)(14,17)(15,18)(16,19)(20,26)(21,24)(22,25)(23,27);(5,8)(6,11)(9,10)(12,14)(13,17)(15,20)(16,21)(18,23)(19,24)(22,27)(25,26);(12,15)(13,18)(14,20)(16,22)(17,23)(19,25)(21,27)(24,26);(5,9)(8,10)(13,19)(17,24)(18,25)(23,26)111 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 11305);