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×#11284 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #397 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| socle | C2×C7 | #1442 |
| prime radical | C24×C7 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C23×C4×C7 ◁ G | |
576 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)(8,9,13,17,14,18,12,27)(10,19,21,15,22,31,28,30)(11,23,26,38,25,16,24,32)(20,34,35,37,36,33,39,29);(8,10)(9,15)(11,20,25,36)(12,28)(13,21)(14,22)(16,29,23,37)(17,31)(18,30)(19,27)(24,39,26,35)(32,34,38,33);(8,11,12,24,14,25,13,26)(9,16,17,32,18,23,27,38)(10,20,21,35,22,36,28,39)(15,29,19,33,30,37,31,34)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4272);