Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11704 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#11946 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#102 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C4×C8 | C23×C7 |
| prime radical | C7×#466 | C4 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ C2×C4×C7×C8 ◁ G | |
302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 55 points:
(1,2,3,4,5,6,7)(8,9,12,17,13,16,11,15)(10,18,19,23,20,14,21,22)(24,25,27,33,41,47,54,49,28,31,42,32,29,34,43,51)(26,35,37,52,44,50,40,45,38,53,36,46,39,48,55,30);(8,10)(9,14)(11,19)(12,21)(13,20)(15,22)(16,18)(17,23)(24,26)(25,30)(27,40)(28,38)(29,44)(31,45)(32,48)(33,50)(34,52)(35,49)(36,43)(37,54)(39,41)(42,55)(46,47)(51,53);(8,11,13,12)(9,15,16,17)(10,19,20,21)(14,22,18,23)(25,31)(30,45)(32,33)(34,47)(35,53)(46,52)(48,50)(49,51)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4934);