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×#60882 |
| 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 | |
1650 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)(11,14)(12,15)(13,18)(17,22)(19,21)(20,23)(24,25)(26,31)(27,33)(28,34)(29,43)(30,35)(32,40)(36,45)(37,46)(38,53)(39,47)(41,48)(42,49)(44,50)(51,54)(52,55);(8,10,12,18)(9,13,15,16)(11,19,20,17)(14,22,23,21)(24,26,30,39)(25,31,35,47)(27,36,41,51)(28,37,29,38)(32,44,49,55)(33,45,48,54)(34,46,43,53)(40,50,42,52);(8,11,12,20)(9,14,15,23)(10,17,18,19)(13,21,16,22)(24,27)(25,32)(26,36)(28,40)(29,42)(30,41)(31,44)(33,43)(34,48)(35,49)(37,50)(38,52)(39,51)(45,53)(46,54)(47,55);(9,15)(13,16)(14,23)(21,22)(24,27)(25,33)(26,36)(28,42)(29,40)(30,41)(31,45)(32,43)(34,49)(35,48)(37,52)(38,50)(39,51)(44,53)(46,55)(47,54);(24,28,30,29)(25,34,35,43)(26,37,39,38)(27,40,41,42)(31,46,47,53)(32,48,49,33)(36,50,51,52)(44,54,55,45)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53870);