Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18446 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C24×C4 |
| direct product | C7×#18465 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C7 |
| prime radical | C7×#538 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
870 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)(11,18)(13,19)(14,21)(16,22)(24,25,30,35)(26,36,39,49)(27,32,41,47)(28,34,29,33)(31,43,45,38)(37,50,53,54)(40,48,42,46)(44,51,55,52);(8,9)(10,13)(11,14)(12,15)(16,20)(17,19)(18,21)(22,23)(24,26,30,39)(25,31,35,45)(27,37,41,53)(28,38,29,43)(32,44,47,55)(33,36,34,49)(40,51,42,52)(46,54,48,50);(8,10)(9,13)(11,16)(12,17)(14,20)(15,19)(18,22)(21,23)(24,27)(25,32)(26,37)(28,42)(29,40)(30,41)(31,44)(33,48)(34,46)(35,47)(36,50)(38,52)(39,53)(43,51)(45,55)(49,54);(8,11,12,18)(9,14,15,21)(10,16,17,22)(13,20,19,23)(24,28,30,29)(25,33,35,34)(26,38,39,43)(27,40,41,42)(31,36,45,49)(32,46,47,48)(37,51,53,52)(44,54,55,50)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 11453);