Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C7 · Schur multiplier (H2) = C28 |
| direct product | C7×#60332 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7×C8 | C24 |
| commutator | C22 | C24×C4×C7 |
| Frattini, cosocle | C2×C4 | C25×C7 |
| socle | C22×C7 | C24×C4 |
| prime radical | C7×#3531 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C7×C8 ◁ 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)(24,25,28,33,29,31,39,44)(26,30,36,42,34,40,47,51)(27,32,37,45,38,43,49,53)(35,41,48,52,46,50,54,55);(8,10,12,18)(9,13,15,16)(11,19,20,17)(14,22,23,21)(24,26)(25,30)(27,35)(28,36)(29,34)(31,40)(32,41)(33,42)(37,48)(38,46)(39,47)(43,50)(44,51)(45,52)(49,54)(53,55);(8,11,12,20)(9,14,15,23)(10,17,18,19)(13,21,16,22)(25,31)(26,34)(32,43)(33,44)(35,46)(36,47)(45,53)(48,54);(9,15)(13,16)(14,23)(21,22)(26,34)(30,40)(35,46)(36,47)(41,50)(42,51)(48,54)(52,55);(24,27,28,37,29,38,39,49)(25,32,33,45,31,43,44,53)(26,35,36,48,34,46,47,54)(30,41,42,52,40,50,51,55)532 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53320);