Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C24×C4 |
| direct product | C7×#10944 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #121 |
| commutator | C2×C4 | C2×C42×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| socle | C22×C7 | #390 |
| prime radical | C22×C7×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C23×C4×C7 ◁ G | |
930 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)(14,23)(15,28)(16,32)(17,21)(18,22)(19,25)(20,29)(24,33)(26,35)(27,34)(30,36)(31,42)(37,39)(38,41)(40,43);(12,14,17,25,18,27,16,26)(13,19,21,34,22,35,32,23)(15,24,30,38,31,40,29,39)(20,33,28,41,36,43,42,37);(9,11)(12,15,16,29,18,31,17,30)(13,20,21,28,22,36,32,42)(14,24,26,39,27,40,25,38)(19,33,34,41,35,43,23,37)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 3932);