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×#61007 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C7 |
| prime radical | C7×#3199 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C4×C7×#41 ◁ G | |
1704 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)(26,30)(27,32)(28,31)(29,34)(33,41)(35,43)(36,42)(37,45)(38,47)(39,48)(40,46)(44,51)(49,53)(50,54)(52,55);(8,10)(9,13)(11,18)(12,17)(14,16)(15,20)(19,23)(21,22)(24,26,29,37)(25,30,34,45)(27,38,39,35)(28,36,41,51)(31,42,33,44)(32,47,48,43)(40,52,53,50)(46,55,49,54);(10,17)(13,16)(18,23)(20,22)(25,31)(28,41)(30,42)(32,46)(33,34)(36,51)(40,53)(43,54)(44,45)(47,55)(48,49)(50,52);(9,14)(13,16)(15,21)(20,22)(24,27)(25,32)(26,35)(28,40)(29,39)(30,43)(31,46)(33,49)(34,48)(36,50)(37,38)(41,53)(42,54)(44,55)(45,47)(51,52);(8,11,12,19)(9,15,14,21)(10,18,17,23)(13,20,16,22)(27,39)(32,48)(35,38)(40,53)(43,47)(46,49)(50,52)(54,55)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53995);