Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C22×C4 |
| direct product | C7×#24948 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C7 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C7 |
| prime radical | C22×C7×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ C2×C4×C7×D4 ◁ G | |
740 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3,4,5,6,7)(8,9)(10,13)(11,12)(14,15)(16,17)(18,20)(19,21)(22,23)(24,25,28,29)(26,30,33,37)(27,34,36,31)(32,35,39,38);(9,12)(13,15)(16,18,19,22)(17,20,21,23)(24,26)(25,30)(27,35)(28,33)(29,37)(31,32)(34,39)(36,38);(8,10,11,14)(9,13,12,15)(17,21)(20,23)(24,27,28,36)(25,31,29,34)(26,32,33,39)(30,38,37,35);(9,12)(13,15)(24,25,28,29)(26,30,33,37)(27,31,36,34)(32,38,39,35)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 17936);