Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60768 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C28 |
| direct product | C7×#61013 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C7 |
| prime radical | C2×C7×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C4×C7×#41 ◁ G | |
1656 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)(26,35,30,31)(27,38,41,46)(28,42,44,33)(32,39,48,36)(37,45,43,49)(40,53,54,51)(47,50,55,52);(8,10)(9,13)(11,18)(12,17)(14,16)(15,20)(19,23)(21,22)(24,26,29,30)(25,31,34,35)(27,39,41,36)(28,43,44,37)(32,46,48,38)(33,49,42,45)(40,50,54,52)(47,51,55,53);(10,17)(13,16)(18,23)(20,22)(24,27,29,41)(25,32,34,48)(26,36,30,39)(28,40,44,54)(31,38,35,46)(33,47,42,55)(37,50,43,52)(45,51,49,53);(9,14)(13,16)(15,21)(20,22)(24,28)(25,33)(26,37)(27,40)(29,44)(30,43)(31,45)(32,47)(34,42)(35,49)(36,50)(38,51)(39,52)(41,54)(46,53)(48,55);(8,11,12,19)(9,15,14,21)(10,18,17,23)(13,20,16,22)(28,44)(33,42)(37,43)(40,54)(45,49)(47,55)(50,52)(51,53)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 54001);