Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9263 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#9276 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #386 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| prime radical | C24×C7 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G | |
522 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)(8,9,12,18,30,37,11,27)(10,21,24,16,32,34,22,38)(13,19,28,15,14,20,31,17)(23,36,25,33,29,35,26,39);(9,15,19,18)(11,28)(12,31)(16,21,35,33)(17,20,27,37)(22,29)(23,24)(34,36,39,38);(8,10,14,26)(9,16,20,36)(11,29,28,22)(12,23,31,24)(13,25,30,32)(15,21,27,39)(17,34,18,33)(19,35,37,38)301 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 2264);