Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13096 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#13102 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #1898 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C2×#41 | C23×C7 |
| prime radical | C7×#3008 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C2×C7×H(5,𝔽2) ◁ G | |
688 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)(8,9,13,16,14,19,12,18)(10,20,24,36,25,38,23,37)(11,26,29,31,28,17,21,34)(15,30,32,27,33,39,35,22);(8,10,21,39)(9,15,31,37)(11,27,12,24)(13,23,28,22)(14,25,29,30)(16,32,26,38)(17,20,18,35)(19,33,34,36);(9,16)(11,28)(12,13)(15,32)(17,34)(18,19)(20,36)(22,27)(23,24)(26,31)(33,35)(37,38)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6090);