Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17597 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#17684 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C42 | C24×C7 |
| prime radical | C7×#3517 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C4×C7×#41 ◁ G | |
932 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,13,18,14,17,12,16)(10,19,20,23,11,15,21,22)(24,25)(26,32,34,39)(27,31)(28,30)(29,33,37,35)(36,38);(8,10,14,11)(9,15,17,19)(12,20,13,21)(16,22,18,23)(24,26)(25,29)(27,35)(28,34)(30,37)(31,39)(32,38)(33,36);(8,11,14,10)(9,15,17,19)(12,20,13,21)(16,23,18,22)(25,30)(29,37)(31,38)(32,39);(8,12,14,13)(9,16,17,18)(10,20,11,21)(15,22,19,23)343 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 10672);