Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1426 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C7 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#7475 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #367 |
| commutator | C16 | C42×C7 |
| Frattini, cosocle | C22×C16 | C22×C7 |
| socle | C2×C7 | #1380 |
| prime radical | C23×C7 | D16 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ C22×C7×C8 ◁ C22×C7×C16 ◁ G | |
170 conjugacy classes of subgroups.
| minimal degree | μ(G) = 71 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 2 generators on 71 points:
(1,2,3,4,5,6,7)(8,9)(10,22,24,50)(11,18)(12,32)(13,35)(14,38)(15,21)(16,23,39,53)(17,29)(19,37)(20,36)(25,58,54,49)(26,60,56,71)(27,62,57,51)(28,45)(30,65)(31,48)(33,43)(34,44)(40,52,67,61)(41,69,68,59)(42,70,66,55)(46,64)(47,63);(8,10,12,25,33,56,63,70,36,57,64,69,37,61,31,53,15,24,29,54,13,26,30,55,14,27,34,59,28,52,11,23)(9,16,18,40,45,68,44,51,38,66,65,71,35,58,17,22,21,39,48,67,19,41,46,62,20,42,47,60,43,49,32,50)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 463);