Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60460 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C27×C4 |
| direct product | C7×#60627 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C7 |
| prime radical | C7×#3499 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C4×C7×#41 ◁ G | |
1998 conjugacy classes of subgroups.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 5 generators on 47 points:
(1,2,3,4,5,6,7)(8,9)(10,13)(11,12)(14,15)(16,17,23,29)(18,25,32,38)(19,24,31,39)(20,26,34,41)(21,28,22,27)(30,40,44,47)(33,37,45,46)(35,43,36,42);(9,12)(13,15)(16,18,21,19,23,32,22,31)(17,24,27,25,29,39,28,38)(20,30,35,33,34,44,36,45)(26,37,42,40,41,46,43,47);(8,10,11,14)(9,13,12,15)(16,19,21,32,23,31,22,18)(17,25,27,39,29,38,28,24)(20,33,35,44,34,45,36,30)(26,40,42,46,41,47,43,37);(16,20)(17,26)(18,30)(19,33)(21,35)(22,36)(23,34)(24,37)(25,40)(27,42)(28,43)(29,41)(31,45)(32,44)(38,47)(39,46);(20,34)(26,41)(30,44)(33,45)(35,36)(37,46)(40,47)(42,43)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53615);