Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18992 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#19013 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C7 |
| prime radical | C7×#3531 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
810 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)(8,9)(10,17)(11,15)(12,20)(13,16)(14,18)(19,22)(21,23)(24,25,30,33)(26,36,39,48)(27,40,42,49)(28,35,29,34)(31,43,46,38)(32,44,47,41)(37,45,53,55)(50,51,54,52);(8,9)(10,14)(11,15)(12,20)(13,16)(17,21)(18,22)(19,23)(24,26,30,39)(25,31,33,46)(27,37,42,53)(28,43,29,38)(32,45,47,55)(34,48,35,36)(40,54,49,50)(41,52,44,51);(8,10)(9,14)(11,18)(12,21)(13,19)(15,22)(16,23)(17,20)(24,27)(25,32)(26,37)(28,41)(29,44)(30,42)(31,45)(33,47)(34,40)(35,49)(36,50)(38,51)(39,53)(43,52)(46,55)(48,54);(8,11,13,12)(9,15,16,20)(10,18,19,21)(14,22,23,17)(25,33)(31,46)(32,47)(34,35)(36,48)(40,49)(45,55)(50,54)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 12001);