Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9385 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×#9386 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #386 |
| commutator | C22×C4 | C22×C3×C4 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| prime radical | C3×#2543 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
618 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(5,8)(6,9)(10,11)(12,13)(14,17,21,26)(15,22,24,18)(16,19)(20,25)(23,27);(4,5,9,11,7,10,6,8)(12,14,16,21)(13,17)(15,23,24,20)(18,27)(19,26)(22,25);(5,10)(8,11)(12,15)(13,18)(14,20)(16,24)(17,25)(19,22)(21,23)(26,27)111 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 2374);