Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19621 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C23×C4 |
| direct product | C7×#19637 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24×C7 |
| prime radical | C2×C7×#127 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
750 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3,4,5,6,7)(8,9)(10,16)(11,19)(12,14)(13,18)(15,20)(17,22)(21,23)(24,25)(26,32)(27,29)(28,36)(30,35)(31,33)(34,38)(37,39);(8,10,11,18,12,17,20,23)(9,13,15,16,14,21,19,22)(25,29)(30,37)(31,36)(32,38);(9,14)(13,21)(15,19)(16,22)(24,26,33,39,27,34,28,35)(25,30,36,38,29,37,31,32);(10,17)(13,21)(16,22)(18,23)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 12625);