Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19050 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#19063 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C7 |
| prime radical | C7×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
806 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 4 generators on 55 points:
(1,2,3,4,5,6,7)(10,16)(11,18)(13,21)(14,19)(17,20)(22,23)(24,25)(26,36)(27,41)(28,33)(29,34)(30,35)(31,39)(32,43)(37,45)(38,46)(40,49)(42,47)(44,48)(50,52)(51,54)(53,55);(8,9,12,15)(10,13,17,22)(11,19,18,14)(16,23,20,21)(24,26)(25,31)(27,37)(28,38)(29,39)(30,40)(32,45)(33,36)(34,49)(35,46)(41,54)(42,51)(43,52)(44,53)(47,55)(48,50);(8,10)(9,13)(11,20)(12,17)(14,23)(15,22)(16,18)(19,21)(24,27,30,44)(25,32,35,47)(26,37,40,53)(28,43,29,42)(31,45,46,55)(33,48,34,41)(36,50,49,54)(38,52,39,51);(24,28,30,29)(25,33,35,34)(26,38,40,39)(27,42,44,43)(31,36,46,49)(32,41,47,48)(37,51,53,52)(45,54,55,50)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 12051);