Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#12567 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #102 |
| commutator | C8 | C2×C42×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C22×C7 | C2×#102 |
| prime radical | C24×C7 | D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C23×C4×C7 ◁ C23×C7×C8 ◁ G | |
586 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)(8,9,10,11)(12,13)(14,25)(15,21)(16,32)(17,22)(18,35)(19,24)(20,27)(23,34)(26,37)(28,36)(29,43)(30,39)(31,40)(33,38)(41,42);(12,14,17,28,18,29,34,27,19,30,31,42,15,26,16,33)(13,20,23,37,21,36,22,39,24,38,32,43,35,41,40,25);(12,15,19,18)(13,21,24,35)(14,26,30,29)(16,31,34,17)(20,36,38,41)(22,32,40,23)(25,37,39,43)(27,28,33,42)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5555);