Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19713 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C23×C8 |
| direct product | C7×#19729 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24×C7 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
788 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)(9,13)(11,14)(15,19)(16,20)(18,21)(22,23)(24,25)(26,34)(27,30)(28,38)(29,33)(31,36)(32,35)(37,39);(8,9,14,20,12,16,11,13)(10,15,21,23,17,22,18,19)(24,26,35,37,29,36,28,27)(25,30,38,31,33,39,32,34);(8,10,12,17)(9,15,16,22)(11,18,14,21)(13,19,20,23)(24,27,28,36,29,37,35,26)(25,31,32,30,33,34,38,39);(10,17)(15,22)(18,21)(19,23)(26,36)(27,37)(30,39)(31,34)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 12717);