Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11867 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#11872 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #1519 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C2×M16 | C23×C7 |
| prime radical | C7×#3060 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C2×C7×#131 ◁ G | |
372 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)(8,9,12,18,14,20,13,19)(10,21,24,16,25,34,22,35)(11,15,28,31,29,32,26,17)(23,36,38,30,27,33,39,37);(9,15)(10,22)(11,26)(12,13)(16,33)(17,19)(18,31)(20,32)(21,30)(24,25)(28,29)(34,37)(35,36)(38,39);(8,10,13,22,14,25,12,24)(9,16,18,34,20,35,19,21)(11,27,28,39,29,23,26,38)(15,30,17,36,32,37,31,33)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4860);