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×#60957 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C7 |
| socle | C22×C7 | C23×D4 |
| prime radical | C7×#3203 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C4×C7×#41 ◁ G | |
1676 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)(11,15)(12,14)(13,17)(18,22)(19,21)(20,23)(24,25,29,33)(26,30,35,44)(27,37,39,47)(28,40,42,32)(31,41,46,38)(34,48,49,55)(36,52,51,45)(43,53,54,50);(8,10)(9,13)(11,18)(12,17)(14,16)(15,20)(19,23)(21,22)(24,26)(25,30)(27,34)(28,36)(29,35)(31,43)(32,45)(33,44)(37,48)(38,50)(39,49)(40,52)(41,53)(42,51)(46,54)(47,55);(10,17)(13,16)(18,23)(20,22)(24,27,29,39)(25,31,33,46)(26,34,35,49)(28,41,42,38)(30,43,44,54)(32,47,40,37)(36,53,51,50)(45,55,52,48);(9,14)(13,16)(15,21)(20,22)(26,35)(30,44)(34,49)(36,51)(43,54)(45,52)(48,55)(50,53);(8,11,12,19)(9,15,14,21)(10,18,17,23)(13,20,16,22)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53945);