Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C22×C4 |
| direct product | C7×#27057 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C7 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C7 |
| prime radical | C2×C7×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ C4×C7×C22⋊C4 ◁ G | |
728 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)(10,14)(11,15)(12,13)(16,17)(18,22)(19,20)(21,23)(24,25,28,31)(26,33,36,30)(27,37,29,32)(34,38,35,39);(9,13)(14,15)(17,20)(21,22)(24,26)(25,30)(27,35)(28,36)(29,34)(31,33)(32,39)(37,38);(8,10,12,11)(9,14,13,15)(16,18,19,23)(17,21,20,22)(25,31)(26,34)(27,29)(30,38)(33,39)(35,36);(8,11,12,10)(9,14,13,15)(17,20)(21,22)(24,27,28,29)(25,32,31,37)(26,35,36,34)(30,39,33,38)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 20045);