Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19713 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C23×C4 |
| direct product | C7×#19732 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24×C7 |
| prime radical | C23×C7 | #127 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
644 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 4 generators on 55 points:
(1,2,3,4,5,6,7)(8,9)(10,18)(11,14)(12,22)(13,17)(15,20)(16,19)(21,23)(24,25,29,33)(26,34,38,48)(27,31,39,45)(28,41,35,32)(30,42,44,37)(36,47,50,55)(40,52,49,46)(43,53,54,51);(8,10,19,21,13,20,12,11)(9,14,22,15,17,23,16,18)(24,26,35,42,29,38,28,37)(25,30,41,48,33,44,32,34)(27,36,49,53,39,50,40,51)(31,43,52,55,45,54,46,47);(8,11,12,20,13,21,19,10)(9,15,16,14,17,18,22,23)(24,27,29,39)(25,31,33,45)(26,36,38,50)(28,40,35,49)(30,43,44,54)(32,46,41,52)(34,47,48,55)(37,51,42,53);(10,20)(11,21)(14,23)(15,18)(27,39)(31,45)(36,50)(40,49)(43,54)(46,52)(47,55)(51,53)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 12720);