Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C26 |
| direct product | C7×#21483 |
| subgroup | G/H | |
|---|---|---|
| center | C7×C8 | #120 |
| commutator | C23 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C2×C7 | C4×#120 |
| prime radical | C7×#3025 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C7×C8 ◁ C22×C7×C8 ◁ G | |
1328 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)(8,9,13,18,14,12,17,26)(10,15,22,29,21,20,28,35)(11,16,25,31,24,23,30,36)(19,32,34,39,33,27,37,38);(8,10)(9,15)(11,19)(12,20)(13,22)(14,21)(16,27)(17,28)(18,29)(23,32)(24,33)(25,34)(26,35)(30,37)(31,38)(36,39);(8,11)(9,16)(10,19)(12,23)(13,25)(14,24)(15,27)(17,30)(18,31)(20,32)(21,33)(22,34)(26,36)(28,37)(29,38)(35,39);(8,12,13,26,14,9,17,18)(10,20,22,35,21,15,28,29)(11,23,25,36,24,16,30,31)(19,32,34,39,33,27,37,38)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 14471);