Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9415 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#9416 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #386 |
| commutator | C42 | C22×C4×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| prime radical | C23×C7 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
380 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)(10,17,18,21)(11,14,19,20)(12,15)(13,16)(22,23)(24,25)(26,35)(27,39,42,49)(28,43)(29,33)(30,34)(31,37)(32,41,48,40)(36,51,53,45)(38,46)(44,47)(50,52,55,54);(8,10,12,20,13,18,22,14)(9,11,15,21,16,19,23,17)(24,26,28,37,30,38,29,44)(25,31,33,35,34,47,43,46)(27,36,41,54,42,53,40,52)(32,45,49,50,48,51,39,55);(8,11)(9,14)(10,15)(12,21)(13,19)(16,20)(17,22)(18,23)(24,27,30,42)(25,32,34,48)(26,36,38,53)(28,41,29,40)(31,45,47,51)(33,49,43,39)(35,50,46,55)(37,54,44,52)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 2404);