Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60771 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C28 |
| direct product | C7×#61019 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C7 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C4×C7×#41 ◁ G | |
1788 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 5 generators on 39 points:
(1,2,3,4,5,6,7)(8,9)(10,16)(11,15)(12,14)(13,17)(18,22)(19,21)(20,23)(25,29)(27,34)(30,35)(31,36)(33,38)(37,39);(8,10)(9,13)(11,18)(12,17)(14,16)(15,20)(19,23)(21,22)(25,30)(26,32)(29,35)(33,38);(10,17)(13,16)(18,23)(20,22)(24,25)(26,31)(27,35)(28,30)(29,34)(32,37)(33,39)(36,38);(9,14)(13,16)(15,21)(20,22)(24,26,28,32)(25,31,30,37)(27,33,34,38)(29,36,35,39);(8,11,12,19)(9,15,14,21)(10,18,17,23)(13,20,16,22)(26,32)(31,37)(33,38)(36,39)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 54007);