Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#13200 |
| subgroup | G/H | |
|---|---|---|
| center | C7×C8 | #127 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C2×C7 | #1955 |
| prime radical | C7×#525 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C7×C8 ◁ C22×C7×C8 ◁ G | |
438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,13,19,14,18,30,34,15,10,16,22,12,21,29,37)(11,23,27,38,28,20,35,36,25,24,31,39,26,17,32,33);(8,10,13,22,14,21,30,37,15,9,16,19,12,18,29,34)(11,24,31,38,28,17,32,36,25,23,27,39,26,20,35,33);(8,11,12,26,15,25,14,28)(9,17,18,23,10,20,21,24)(13,27,29,32,16,31,30,35)(19,33,34,38,22,36,37,39)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6188);