Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#13186 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #476 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C23×C7 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C4×C7×C8 ◁ G | |
310 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4,5,6,7)(9,11)(13,19,22,20)(14,23)(15,24)(16,17)(21,27,26,25);(12,13,17,26,18,22,16,21)(14,20,15,25,24,19,23,27);(8,9,10,11)(12,14,16,23,18,24,17,15)(13,20,21,27,22,19,26,25)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6174);