Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #22455 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C26 |
| direct product | C7×#22456 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | D4×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| prime radical | C2×C7×#577 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ G | |
1336 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)(10,18)(11,23,25,15)(12,26)(13,17)(14,21)(16,27)(19,33,36,32)(20,30)(22,29)(24,31,39,38)(28,35,37,34);(8,10,13,22)(9,14,17,30)(11,19,25,36)(12,20,27,21)(15,28,23,37)(16,29,26,18)(24,34,39,35)(31,33,38,32);(8,11)(9,15)(10,19)(12,24)(13,25)(14,28)(16,31)(17,23)(18,32)(20,34)(21,35)(22,36)(26,38)(27,39)(29,33)(30,37);(10,20)(12,27)(14,29)(16,26)(18,30)(19,34)(21,22)(24,39)(28,33)(31,38)(32,37)(35,36)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 15444);