Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#12032 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C2×D8 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C4×C8 | C23×C7 |
| socle | C22×C7 | C2×#107 |
| prime radical | C23×C7 | #107 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ C2×C4×C7×C8 ◁ G | |
222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,11,14,12,15,18,21)(10,13,17,20,16,19,22,23)(25,29)(26,30)(27,33)(31,39)(32,37)(34,38)(35,36);(8,10)(9,13)(11,17)(12,16)(14,20)(15,19)(18,22)(21,23)(24,25,30,36,27,31,34,37,28,32,38,39,33,35,26,29);(10,16)(13,19)(17,22)(20,23)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5020);