Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60768 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C28 |
| direct product | C7×#61010 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C7 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C4×C7×#41 ◁ G | |
1700 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)(11,15)(12,14)(13,17)(18,22)(19,21)(20,23)(24,25)(26,35)(27,39)(28,33)(29,31)(30,36)(32,41)(34,43)(37,45)(38,44)(40,48)(42,49)(46,50)(47,53)(51,55)(52,54);(8,10)(9,13)(11,18)(12,17)(14,16)(15,20)(19,23)(21,22)(24,26)(25,30)(27,40)(28,36)(29,38)(31,46)(32,48)(33,35)(34,44)(37,41)(39,45)(42,52)(43,50)(47,54)(49,55)(51,53);(10,17)(13,16)(18,23)(20,22)(25,31)(26,36)(29,43)(30,44)(32,47)(33,34)(35,46)(37,40)(39,49)(42,53)(45,54)(48,55);(9,14)(13,16)(15,21)(20,22)(24,27)(25,32)(26,37)(28,41)(29,42)(30,45)(31,47)(33,39)(34,49)(35,48)(36,40)(38,51)(43,53)(44,54)(46,55)(50,52);(8,11,12,19)(9,15,14,21)(10,18,17,23)(13,20,16,22)(27,41)(32,39)(37,40)(42,53)(45,48)(47,49)(51,52)(54,55)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53998);