Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C8 · Schur multiplier (H2) = C24 |
| direct product | C7×#12981 |
| subgroup | G/H | |
|---|---|---|
| center | C7×C8 | #120 |
| commutator | C23 | C22×C7×C8 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C2×C7 | #1869 |
| prime radical | C7×#525 | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C7×C8 ◁ C22×C7×C8 ◁ G | |
488 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)(8,9,12,17,13,18,26,32,14,10,15,20,27,21,28,34)(11,19,23,31,24,33,36,39,22,16,25,29,35,30,37,38);(8,10,12,20,13,21,26,34,14,9,15,17,27,18,28,32)(11,19,25,29,24,33,37,38,22,16,23,31,35,30,36,39);(8,11)(9,16)(10,19)(12,25)(13,24)(14,22)(15,23)(17,31)(18,30)(20,29)(21,33)(26,37)(27,35)(28,36)(32,39)(34,38)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5969);