Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2190 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#13024 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #447 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C2×C7 | #1879 |
| prime radical | C7×#3414 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ C2×C7×#97 ◁ G | |
466 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)(8,9,14,20,15,21,13,32,16,22,33,39,31,38,12,19)(10,23,24,36,25,37,11,27,26,18,29,17,28,34,30,35);(8,10,15,28,16,26,31,25)(9,17,21,27,22,36,38,35)(11,13,24,33,30,12,29,14)(18,19,37,20,23,32,34,39);(8,11)(9,18)(10,13)(12,26)(14,25)(15,24)(16,30)(17,19)(20,27)(21,37)(22,23)(28,33)(29,31)(32,36)(34,38)(35,39)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6012);