Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C4 |
| direct product | C7×#13118 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #442 |
| 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×C7×#97 ◁ G | |
272 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3,4,5,6,7)(8,9)(10,13)(11,12)(14,15)(17,20)(18,26,28,23)(19,21)(24,25)(27,31,29,30);(9,12)(13,15)(16,17,21,24,22,25,19,20)(18,23,27,31,28,26,29,30);(8,10,11,14)(9,13,12,15)(16,18)(17,23)(19,29)(20,30)(21,27)(22,28)(24,31)(25,26)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6106);