Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C7×#13196 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7×C8 | C2×D4 |
| commutator | C2×C4 | C22×C7×C8 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C22×C7 | C8×D4 |
| prime radical | C2×C7×#135 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7×C8 ◁ C22×C7×C16 ◁ G | |
430 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)(12,13,18,24,19,23,34,40,16,22,26,21,17,35,15,32)(14,25,30,39,31,41,27,42,28,20,36,37,29,38,33,43);(8,9,10,11)(12,14,26,36,19,31,15,33,16,28,18,30,17,29,34,27)(13,20,21,39,23,38,32,42,22,25,24,37,35,41,40,43);(12,15,17,26,16,34,19,18)(13,21,23,32,22,24,35,40)(14,27,29,30,28,33,31,36)(20,37,38,43,25,39,41,42)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6184);