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×#22459 |
| 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 | |
1312 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)(8,9,13,18,14,19,12,27)(10,20,23,15,24,33,28,32)(11,17,25,34,26,16,29,35)(21,37,22,30,39,31,38,36);(8,10,14,24)(9,15,19,32)(11,21,26,39)(12,28,13,23)(16,30,17,36)(18,33,27,20)(22,29,38,25)(31,34,37,35);(8,11)(9,16)(10,21)(12,25)(13,29)(14,26)(15,30)(17,19)(18,34)(20,31)(22,28)(23,38)(24,39)(27,35)(32,36)(33,37);(8,11)(9,17)(10,22)(12,29)(13,25)(14,26)(15,31)(16,19)(18,34)(20,30)(21,28)(23,39)(24,38)(27,35)(32,37)(33,36)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 15447);