Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C7×C8 · Schur multiplier (H2) = C22 |
| direct product | C7×#7181 |
| subgroup | G/H | |
|---|---|---|
| center | C7×C8 | #102 |
| commutator | C8 | C4×C7×C8 |
| Frattini, cosocle | C2×C4×C8 | C22×C7 |
| socle | C2×C7 | #1279 |
| prime radical | C23×C7 | #104 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C7×C8 ◁ C22×C7×C8 ◁ C2×C4×C7×C8 ◁ G | |
240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 2 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,12,17,14,19,30,35,15,20,29,36,13,18,11,27)(10,21,23,32,25,38,22,16,26,31,39,33,24,37,28,34);(8,10,13,24,15,26,14,25)(9,16)(11,28,29,39,30,22,12,23)(17,31)(18,32)(19,33)(20,34)(21,36)(27,38)(35,37)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 169);