Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#27067 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C7 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C7 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ C4×C7×C4⋊C4 ◁ G | |
700 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 1792 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,12,13)(10,14,11,15)(16,17,19,20)(18,22,23,21)(24,25)(26,29)(27,30)(28,31)(32,39)(33,37)(34,36)(35,38);(9,13)(14,15)(17,20)(21,22)(24,26,27,33)(25,29,30,37)(28,32,35,34)(31,39,38,36);(8,10,12,11)(9,14,13,15)(16,18,19,23)(17,21,20,22)(25,30)(26,32)(28,35)(29,36)(33,34)(37,39);(8,11,12,10)(9,14,13,15)(17,20)(21,22)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 20055);