Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#11302 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #397 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| socle | C2×C7 | #1443 |
| prime radical | C24×C7 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C23×C4×C7 ◁ G | |
532 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)(8,9,13,19,14,17,12,18)(10,15,23,32,21,31,22,30)(11,24,26,33,27,34,28,16)(20,35,36,39,25,38,37,29);(8,10,12,23,14,21,13,22)(9,15,19,30,17,31,18,32)(11,25,26,36,27,20,28,37)(16,29,34,35,33,39,24,38);(8,11)(9,16,17,33)(10,20,21,25)(12,26)(13,28)(14,27)(15,29)(18,24,19,34)(22,36,23,37)(30,35)(31,39)(32,38)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4290);