Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18284 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#18315 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C7 |
| prime radical | C7×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
868 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 55 points:
(1,2,3,4,5,6,7)(8,9)(10,17)(11,15)(12,20)(13,16)(14,18)(19,22)(21,23)(24,25,29,33)(26,30,36,43)(27,31,38,45)(28,39,40,32)(34,46,48,41)(35,49,50,42)(37,51,52,44)(47,53,55,54);(8,10)(9,14)(11,18)(12,21)(13,19)(15,22)(16,23)(17,20)(24,26)(25,30)(27,34)(28,35)(29,36)(31,41)(32,42)(33,43)(37,47)(38,48)(39,49)(40,50)(44,53)(45,46)(51,54)(52,55);(8,11,13,12)(9,15,16,20)(10,18,19,21)(14,22,23,17)(24,27)(25,31)(26,34)(28,37)(29,38)(30,41)(32,44)(33,45)(35,47)(36,48)(39,51)(40,52)(42,53)(43,46)(49,54)(50,55);(8,11,13,12)(9,15,16,20)(10,18,19,21)(14,22,23,17)(24,28,29,40)(25,32,33,39)(26,35,36,50)(27,37,38,52)(30,42,43,49)(31,44,45,51)(34,47,48,55)(41,53,46,54)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 11303);