Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C28 |
| direct product | C7×#60859 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C7 |
| socle | C22×C7 | C23×D4 |
| prime radical | C7×#3455 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C7×#117 ◁ G | |
1654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 448 |
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)(27,39)(28,33)(29,42)(30,34)(32,40)(35,49)(36,45)(37,46)(38,52)(41,47)(43,48)(44,51)(50,55)(53,54);(8,10,12,18)(9,13,15,16)(11,19,20,17)(14,22,23,21)(24,26)(25,31)(27,35)(28,37)(29,38)(30,36)(32,44)(33,46)(34,45)(39,49)(40,51)(41,50)(42,52)(43,53)(47,55)(48,54);(8,11,12,20)(9,14,15,23)(10,17,18,19)(13,21,16,22)(24,27)(25,32)(26,35)(28,40)(29,43)(30,41)(31,44)(33,47)(34,48)(36,50)(37,51)(38,53)(39,42)(45,54)(46,55)(49,52);(9,15)(13,16)(14,23)(21,22)(26,36)(31,45)(35,50)(37,38)(44,54)(46,52)(49,55)(51,53);(24,28,30,29)(25,33,34,42)(26,37,36,38)(27,40,41,43)(31,46,45,52)(32,47,48,39)(35,51,50,53)(44,55,54,49)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53847);