Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#13180 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #459 |
| 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 | |
294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7)(8,9,11,10)(12,13)(14,24)(15,21)(16,31)(17,34)(18,23)(19,25)(20,29)(22,33)(26,39)(27,41)(28,35)(30,40)(32,36)(37,42)(38,43);(8,10,11,9)(12,14)(13,19)(15,25)(16,32)(17,26)(18,28)(20,35)(21,39)(22,36)(23,38)(24,31)(27,33)(29,42)(30,43)(34,37)(40,41);(12,15,17,29,18,30,33,16)(13,20,22,21,23,31,34,40)(14,25,26,42,28,43,27,32)(19,35,36,39,38,24,37,41)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6168);