Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10192 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#10196 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23×C3 |
| prime radical | C3×#3245 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
546 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)(4,5,8,6)(7,10,9,11)(12,13,17,21)(14,22,24,27)(15,20,16,19)(18,25,26,23);(5,10)(6,11)(7,9)(12,14)(13,18)(15,25)(16,23)(17,24)(19,27)(20,22)(21,26);(4,6)(5,8)(7,11)(9,10)(12,14)(13,18)(15,23)(16,25)(17,24)(19,22)(20,27)(21,26)111 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3184);