Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23400 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C27 |
| direct product | C7×#23401 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C7 |
| prime radical | C7×#3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G | |
1954 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)(10,11)(12,13)(14,22)(15,19)(16,28)(17,21)(18,24)(20,31)(23,39)(25,37)(26,35)(27,38)(29,33)(30,36)(32,42)(34,41)(40,43);(12,14,17,25)(13,18,21,33)(15,23,26,40)(16,29,31,24)(19,34,35,32)(20,37,28,22)(27,42,30,41)(36,43,38,39);(9,11)(12,15)(13,19)(14,23)(16,30)(17,26)(18,32)(20,38)(21,35)(22,39)(24,42)(25,40)(27,31)(28,36)(29,41)(33,34)(37,43);(15,26)(19,35)(23,40)(27,30)(32,34)(36,38)(39,43)(41,42)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 16389);