Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#25190 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C7 |
| prime radical | C7×#3013 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ G | |
992 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3,4,5,6,7)(12,13,17,23,19,16,18,22)(14,24,26,21,15,27,25,20);(8,9)(10,11)(12,14,17,26,19,15,18,25)(13,20,23,24,16,21,22,27);(9,11)(12,15,19,14)(13,21,16,20)(17,26,18,25)(22,27,23,24);(12,16)(13,19)(14,21)(15,20)(17,22)(18,23)(24,25)(26,27)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 18178);