Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33359 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C26 |
| direct product | C7×#33370 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | C22×#99 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C2×Q8 | C24×C7 |
| prime radical | C7×#3500 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C7×D4 ◁ C2×C7×2−1+4 ◁ G | |
1208 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,12,17,13,18,25,32)(10,15,21,30,19,28,34,38)(11,16,23,31,22,14,24,27)(20,29,35,39,33,26,36,37);(9,14)(11,22)(15,26)(16,18)(17,31)(20,33)(23,24)(27,32)(28,29)(30,39)(35,36)(37,38);(8,10)(9,15)(11,20)(12,21)(13,19)(14,26)(16,29)(17,30)(18,28)(22,33)(23,35)(24,36)(25,34)(27,37)(31,39)(32,38);(10,19)(15,28)(20,33)(21,34)(26,29)(30,38)(35,36)(37,39)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 26358);