Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11115 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#11266 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #388 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| socle | C2×C7 | #1442 |
| prime radical | C24×C7 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C23×C4×C7 ◁ G | |
560 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)(8,9,13,17,14,19,12,18)(10,20,22,15,24,31,23,32)(11,25,27,35,28,34,29,16)(21,37,38,33,39,30,26,36);(8,10,12,23,14,24,13,22)(9,15,18,20,19,32,17,31)(11,26,27,21,28,38,29,39)(16,33,25,30,35,36,34,37);(8,11,12,27,14,28,13,29)(9,16,17,34,19,35,18,25)(10,21,22,26,24,39,23,38)(15,30,20,36,32,37,31,33)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4254);