Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61629 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C28 |
| direct product | C7×#61867 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C7 |
| prime radical | C7×#3119 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C7×#533 ◁ G | |
1628 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 5 generators on 55 points:
(1,2,3,4,5,6,7)(8,9)(10,16)(11,14)(12,15)(13,18)(17,22)(19,21)(20,23)(24,25)(26,31)(27,38)(28,33)(29,42)(30,34)(32,39)(35,50)(36,45)(37,46)(40,48)(41,47)(43,49)(44,53)(51,55)(52,54);(8,10,12,18)(9,13,15,16)(11,19,20,17)(14,22,23,21)(24,26)(25,31)(27,35)(28,41)(29,37)(30,36)(32,44)(33,47)(34,45)(38,50)(39,53)(40,51)(42,46)(43,52)(48,55)(49,54);(24,27)(25,32)(26,35)(28,39)(29,43)(30,40)(31,44)(33,48)(34,49)(36,51)(37,52)(38,42)(41,53)(45,54)(46,50)(47,55);(8,11,12,20)(9,14,15,23)(10,17,18,19)(13,21,16,22)(26,36)(31,45)(35,51)(37,41)(44,54)(46,47)(50,55)(52,53);(9,15)(13,16)(14,23)(21,22)(24,28,30,29)(25,33,34,42)(26,37,36,41)(27,39,40,43)(31,46,45,47)(32,48,49,38)(35,52,51,53)(44,50,54,55)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 54855);