Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C26 |
| direct product | C7×#22462 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | D42 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C2×C7 | D4×C22⋊C4 |
| prime radical | C7×#3584 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ G | |
1320 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)(8,9,14,18)(10,19,23,21)(11,24)(12,27,13,17)(15,28,31,22)(16,29)(20,35)(25,33)(26,32)(30,39)(34,37)(36,38);(8,10)(9,15)(11,20)(12,28)(13,22)(14,23)(16,30)(17,21)(18,31)(19,27)(24,34)(25,38)(26,39)(29,37)(32,36)(33,35);(8,11)(9,16)(10,20)(12,25)(13,29)(14,26)(15,30)(17,32)(18,33)(19,34)(21,36)(22,37)(23,39)(24,27)(28,38)(31,35);(8,9)(10,21)(11,16)(12,27)(13,17)(14,18)(15,28)(19,23)(20,36)(22,31)(24,25)(26,33)(29,32)(30,38)(34,39)(35,37)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 15450);