Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#11320 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #124 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| socle | C22×C7 | #397 |
| prime radical | C24×C7 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C23×C4×C7 ◁ G | |
546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7)(9,10)(12,13)(14,22,25,35)(15,19)(16,20)(17,21)(18,24,32,29)(23,38,26,37)(27,33)(28,34)(30,36)(31,41,42,40)(39,43);(8,9)(10,11)(12,14,16,29,17,25,30,24)(13,18,20,35,21,32,36,22)(15,26,27,40,28,23,39,41)(19,31,33,38,34,42,43,37);(8,10,11,9)(12,15,16,27,17,28,30,39)(13,19,20,33,21,34,36,43)(14,23,24,40,25,26,29,41)(18,31,22,37,32,42,35,38)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4308);