Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11858 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#11866 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#102 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C7×#466 | C22 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ C2×C4×C7×C8 ◁ G | |
306 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)(10,16,18,13)(11,14)(12,15)(17,22,19,21)(20,23)(24,25,29,35,31,37,30,36)(26,38,41,53,43,54,42,49)(27,44,28,47,46,34,45,33)(32,48,50,40,52,39,51,55);(8,10)(9,13)(11,19)(12,18)(14,22)(15,16)(17,20)(21,23)(24,26)(25,32)(27,40)(28,48)(29,42)(30,41)(31,43)(33,38)(34,53)(35,51)(36,50)(37,52)(39,45)(44,49)(46,55)(47,54);(8,11,12,20)(9,14,15,23)(10,17,18,19)(13,21,16,22)(24,27,30,45,31,46,29,28)(25,33,35,44,37,47,36,34)(26,39,42,40,43,48,41,55)(32,49,50,38,52,53,51,54)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4854);