Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C7 · Schur multiplier (H2) = C213 |
| direct product | C22×C7×#540 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C7 | C24 |
| commutator, Frattini, cosocle | C22 | C26×C7 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C7 ◁ G | |
8838 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 6 generators on 27 points:
(1,2)(21,22,23,24,25,26,27);(3,4);(5,6)(7,11)(8,9)(10,12)(14,17)(15,19);(6,9)(10,11)(13,14,16,17)(15,18,19,20);(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20);(5,7,8,12)(6,10,9,11)(13,15,16,19)(14,18,17,20)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 55665);