Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#11273 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #394 |
| 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)(10,14,19,29)(11,21)(12,25)(13,16)(15,23)(17,26)(18,32,22,37)(20,36,35,28)(24,38)(27,39,30,34)(31,33);(8,10,12,20,13,19,26,35)(9,14,17,28,16,29,25,36)(11,22,23,39,24,18,33,34)(15,30,21,37,31,27,38,32);(8,11)(9,15,16,31)(10,18,19,22)(12,23)(13,24)(14,27)(17,21,25,38)(20,34,35,39)(26,33)(28,32)(29,30)(36,37)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4261);