Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#13039 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #442 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C2×C7×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C7×#97 ◁ G | |
472 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)(9,11)(12,13,18,24,28,43,16,35,20,26,17,37,19,25,36,23)(14,27,29,41,38,22,34,40,31,42,33,21,30,39,15,32);(8,9,10,11)(12,14,28,30,20,31,19,38)(13,21,25,32,26,41,43,40)(15,16,33,17,34,36,29,18)(22,24,42,23,39,37,27,35);(12,15)(13,22)(14,16)(17,30)(18,38)(19,29)(20,34)(21,24)(23,32)(25,42)(26,39)(27,43)(28,33)(31,36)(35,40)(37,41)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6027);