Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#18082 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C22×D4 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C42 | C24×C7 |
| socle | C22×C7 | C2×C4×D4 |
| prime radical | C7×#3171 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C7×#431 ◁ G | |
612 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)(8,9,14,18)(10,19,11,16)(12,17,13,22)(15,21,23,20)(24,25,29,32,28,31,37,39)(26,33,35,36,34,30,38,27);(8,10,14,11)(9,15,18,23)(12,20,13,21)(16,22,19,17)(24,26)(25,30)(27,32)(28,34)(29,35)(31,33)(36,39)(37,38);(8,11,14,10)(9,16,18,19)(12,20,13,21)(15,22,23,17)(24,27,29,33,28,36,37,30)(25,26,32,35,31,34,39,38);(8,12,14,13)(9,17,18,22)(10,20,11,21)(15,16,23,19)(26,34)(27,36)(30,33)(35,38)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 11070);