Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#21184 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | C2×#121 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C2×C7 | C2×#389 |
| prime radical | C24×C7 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C23×C4×C7 ◁ G | |
844 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,13,17,14,18,12,28)(10,19,21,15,23,30,22,31)(11,24,27,32,25,38,26,16)(20,29,35,37,34,39,36,33);(8,10,13,21,14,23,12,22)(9,15,17,30,18,31,28,19)(11,20,27,35,25,34,26,36)(16,29,24,37,32,39,38,33);(8,11,12,26,14,25,13,27)(9,16,17,24,18,32,28,38)(10,20,22,36,23,34,21,35)(15,29,30,37,31,39,19,33);(11,25)(16,32)(20,34)(24,38)(26,27)(29,39)(33,37)(35,36)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 14172);