Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C8 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#13124 |
| 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×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7×C8 ◁ C22×C7×C16 ◁ G | |
332 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)(12,13,18,24,17,22,15,33,19,25,34,21,16,23,27,40)(14,26,31,42,30,39,28,36,32,37,43,41,29,20,35,38);(12,14,27,35,16,29,34,43,19,32,15,28,17,30,18,31)(13,20,33,42,23,37,24,38,25,39,40,41,22,26,21,36);(9,11)(12,15,16,18,19,27,17,34)(13,21,22,40,25,24,23,33)(14,28,29,31,32,35,30,43)(20,36,26,41,39,38,37,42)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6112);