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×#61020 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C7 |
| prime radical | C7×#3512 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C4×C7×#41 ◁ G | |
1668 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 5 generators on 55 points:
(1,2,3,4,5,6,7)(8,9)(10,16)(11,15)(12,14)(13,17)(18,22)(19,21)(20,23)(24,25,29,34)(26,30,35,44)(27,32,39,48)(28,31,41,33)(36,43,38,47)(37,45,50,42)(40,46,53,49)(51,55,52,54);(8,10)(9,13)(11,18)(12,17)(14,16)(15,20)(19,23)(21,22)(24,26)(25,30)(27,38)(28,37)(29,35)(31,45)(32,47)(33,42)(34,44)(36,39)(40,52)(41,50)(43,48)(46,54)(49,55)(51,53);(10,17)(13,16)(18,23)(20,22)(25,31)(26,35)(28,41)(30,42)(32,46)(33,34)(36,38)(40,53)(43,54)(44,45)(47,55)(48,49);(9,14)(13,16)(15,21)(20,22)(24,27,29,39)(25,32,34,48)(26,36,35,38)(28,40,41,53)(30,43,44,47)(31,46,33,49)(37,51,50,52)(42,54,45,55);(8,11,12,19)(9,15,14,21)(10,18,17,23)(13,20,16,22)(27,39)(32,48)(36,38)(40,53)(43,47)(46,49)(51,52)(54,55)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 54008);