Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C26 |
| direct product | C7×#20678 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #121 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C22×C7 | C2×#121 |
| prime radical | C22×C7×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C23×C4×C7 ◁ G | |
1330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3,4,5,6,7)(8,9)(10,11)(12,13,17,22,18,23,16,21)(14,24,27,35,28,34,26,19)(15,29,30,20,32,36,31,37)(25,39,40,38,42,33,41,43);(12,14,16,26,18,28,17,27)(13,19,22,24,23,35,21,34)(15,25,30,40,32,42,31,41)(20,33,29,38,37,39,36,43);(9,11)(12,15)(13,20)(14,25)(16,30)(17,31)(18,32)(19,33)(21,36)(22,29)(23,37)(24,38)(26,40)(27,41)(28,42)(34,43)(35,39);(12,16,18,17)(13,21,23,22)(14,26,28,27)(15,30,32,31)(19,34,35,24)(20,36,37,29)(25,40,42,41)(33,43,39,38)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 13666);