Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#13182 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #459 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C2×C7 | #1947 |
| prime radical | C23×C7 | #137 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ C2×C4×C7×C8 ◁ G | |
280 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)(8,9,14,19,15,20,13,30)(10,21,24,37,25,31,23,35)(11,18,12,28,27,36,26,17)(16,29,32,39,34,38,33,22);(8,10,14,24,15,25,13,23)(9,16,19,32,20,34,30,33)(11,22,12,29,27,39,26,38)(17,31,18,35,28,21,36,37);(8,11,13,26,15,27,14,12)(9,17,19,18,20,28,30,36)(10,22,23,38,25,39,24,29)(16,31,32,35,34,21,33,37)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6170);