Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2238 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C4 |
| direct product | C7×#13115 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #481 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C2×C7 | #1919 |
| prime radical | C23×C7 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ C2×C7×#97 ◁ G | |
252 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)(8,9,14,19,12,18,15,29)(10,20,23,37,22,16,13,27)(11,24,25,31,26,39,30,36)(17,32,33,38,35,28,34,21);(8,10,14,23,12,22,15,13)(9,16,19,27,18,20,29,37)(11,21,25,32,26,38,30,28)(17,31,33,39,35,36,34,24);(8,11,14,25,12,26,15,30)(9,17,29,34,18,35,19,33)(10,21,23,32,22,38,13,28)(16,31,37,24,20,36,27,39)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6103);