Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#11306 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #124 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| socle | C22×C7 | #400 |
| prime radical | C24×C7 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C23×C4×C7 ◁ G | |
530 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)(12,13,18,21,17,22,16,23)(14,24,28,34,27,35,26,19)(15,20,31,37,32,38,30,39)(25,36,29,42,41,43,40,33);(8,9)(10,11)(12,14,18,28,17,27,16,26)(13,19,23,35,22,34,21,24)(15,29,30,25,32,40,31,41)(20,36,37,42,38,43,39,33);(9,11)(12,15,16,30,17,32,18,31)(13,20,21,37,22,38,23,39)(14,25,26,40,27,41,28,29)(19,33,24,36,34,42,35,43)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4294);