Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19733 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C23×C4 |
| direct product | C7×#19740 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24×C7 |
| prime radical | C7×#3207 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
704 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)(10,17)(11,15)(12,21)(13,16)(14,18)(19,22)(20,23)(26,32)(27,34)(29,35)(31,37)(33,38)(36,39);(8,10,11,20,13,19,12,18)(9,14,21,22,16,23,15,17)(24,25)(26,29)(27,31)(28,30)(32,35)(33,36)(34,37)(38,39);(8,9)(10,14)(11,21)(12,15)(13,16)(17,18)(19,23)(20,22)(24,26,27,32,28,33,34,38)(25,29,31,35,30,36,37,39);(8,11,13,12)(9,15,16,21)(10,18,19,20)(14,22,23,17)(25,30)(29,36)(31,37)(35,39)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 12728);