Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9369 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#9374 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #386 |
| commutator | C42 | C22×C4×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| prime radical | C23×C7 | #123 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
368 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 55 points:
(1,2,3,4,5,6,7)(8,9,14,17,12,11,13,16)(10,18,20,22,21,15,19,23)(24,25,29,33,30,34,28,44)(26,35,39,46,38,50,37,31)(27,40,42,48,43,32,41,47)(36,49,53,45,52,55,51,54);(8,10,14,19,12,21,13,20)(9,15,16,23,11,18,17,22)(24,26,30,38)(25,31)(27,36,43,52)(28,37,29,39)(32,45)(33,35)(34,46)(40,54)(41,51,42,53)(44,50)(47,55)(48,49);(8,11,12,9)(10,18,21,15)(13,16,14,17)(19,22,20,23)(24,27,30,43)(25,32,34,40)(26,36,38,52)(28,42,29,41)(31,45,46,54)(33,48,44,47)(35,49,50,55)(37,53,39,51)259 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 2362);