Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C7 · Schur multiplier (H2) = C29 |
| direct product | C4×C7×#578 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C22×D4 |
| commutator | C4 | C24×C4×C7 |
| Frattini, cosocle | C2×C4 | C25×C7 |
| socle | C22×C7 | C23×D4 |
| prime radical | C2×C7×#578 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C4×C7×#41 ◁ G | |
2658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 5 generators on 43 points:
(1,2,3,4)(37,38,39,40,41,42,43);(5,6,10,14)(7,15,18,28)(8,12,19,26)(9,21,23,13)(11,22,25,17)(16,29,30,36)(20,33,32,27)(24,34,35,31);(5,7,10,18)(6,11,14,25)(8,16,19,30)(9,22,23,17)(12,24,26,35)(13,28,21,15)(20,34,32,31)(27,36,33,29);(5,8)(6,12)(7,16)(9,20)(10,19)(11,24)(13,27)(14,26)(15,29)(17,31)(18,30)(21,33)(22,34)(23,32)(25,35)(28,36);(8,19)(12,26)(16,30)(20,32)(24,35)(27,33)(29,36)(31,34)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53161);