Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8479 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#8489 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #379 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| prime radical | C7×#3560 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
690 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 55 points:
(1,2,3,4,5,6,7)(8,9)(10,17,19,14)(11,13)(12,16)(15,20)(18,22,23,21)(24,25,28,33,29,32,43,47)(26,34,37,48,36,30,41,45)(27,38,40,49,39,31,42,46)(35,50,52,55,51,44,53,54);(9,13)(11,20)(14,21)(15,16)(17,22)(18,23)(24,26)(25,30)(27,35)(28,41)(29,36)(31,44)(32,34)(33,48)(37,43)(38,50)(39,51)(40,53)(42,52)(45,47)(46,55)(49,54);(8,10)(9,14)(11,18)(12,19)(13,21)(15,22)(16,17)(20,23)(24,27)(25,31)(26,35)(28,42)(29,39)(30,44)(32,38)(33,49)(34,50)(36,51)(37,53)(40,43)(41,52)(45,55)(46,47)(48,54)259 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 1477);