Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60753 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C28 |
| direct product | C7×#60869 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C7 |
| prime radical | C24×C7 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C7×#117 ◁ G | |
1626 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)(24,25,30,35)(26,31,37,45)(27,40,42,49)(28,34,29,33)(32,43,48,41)(36,50,51,55)(38,47,39,46)(44,53,54,52);(8,10,12,18)(9,13,15,16)(11,19,20,17)(14,22,23,21)(24,26)(25,31)(27,36)(28,38)(29,39)(30,37)(32,44)(33,46)(34,47)(35,45)(40,50)(41,52)(42,51)(43,53)(48,54)(49,55);(8,11,12,20)(9,14,15,23)(10,17,18,19)(13,21,16,22)(24,27)(25,32)(26,36)(28,41)(29,43)(30,42)(31,44)(33,40)(34,49)(35,48)(37,51)(38,52)(39,53)(45,54)(46,50)(47,55);(9,15)(13,16)(14,23)(21,22)(26,37)(31,45)(36,51)(38,39)(44,54)(46,47)(50,55)(52,53);(24,28,30,29)(25,33,35,34)(26,38,37,39)(27,41,42,43)(31,46,45,47)(32,40,48,49)(36,52,51,53)(44,50,54,55)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53857);