Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#27023 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C7 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C7 |
| prime radical | C22×C7×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ C4×C7×C4⋊C4 ◁ G | |
690 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)(8,9,12,15)(10,14,11,13)(16,17,19,21)(18,22,23,20)(24,25,29,33)(26,30,35,38)(27,31,28,32)(34,37,36,39);(8,10,12,11)(9,13,15,14)(16,18,19,23)(17,20,21,22)(24,26)(25,30)(27,34)(28,36)(29,35)(31,37)(32,39)(33,38);(17,21)(20,22)(24,27)(25,31)(28,29)(32,33)(34,36)(37,39);(8,11,12,10)(9,14,15,13)(17,21)(20,22)(24,27,29,28)(25,31,33,32)(26,34,35,36)(30,37,38,39)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 20011);