Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#21181 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #121 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C22×C7 | C2×#121 |
| prime radical | C24×C7 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C23×C4×C7 ◁ G | |
874 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3,4,5,6,7)(8,9,11,10)(12,13)(14,24)(15,29)(16,21)(17,22)(18,23)(19,27)(20,31)(25,39)(26,34)(28,37)(30,36)(32,35)(33,42)(38,41)(40,43);(8,10,11,9)(12,14)(13,19)(15,25)(16,26)(17,27)(18,28)(20,33)(21,24)(22,37)(23,34)(29,38)(30,40)(31,41)(32,42)(35,43)(36,39);(12,15,17,31,18,32,16,30)(13,20,21,29,23,36,22,35)(14,25,27,41,28,42,26,40)(19,33,24,38,34,39,37,43);(12,16,18,17)(13,21,23,22)(14,26,28,27)(15,30,32,31)(19,24,34,37)(20,29,36,35)(25,40,42,41)(33,38,39,43)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 14169);