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) = C25 |
| direct product | C7×#22694 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | D42 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C2×C7 | D4×C4⋊C4 |
| prime radical | C2×C7×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ G | |
812 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)(8,9,14,19,15,20,13,32)(10,21,23,16,25,34,24,35)(11,26,22,36,28,33,27,17)(12,18,29,38,31,39,30,37);(8,10)(9,16)(11,12)(13,24)(14,23)(15,25)(17,18)(19,34)(20,35)(21,32)(22,29)(26,38)(27,30)(28,31)(33,37)(36,39);(8,11,13,27,15,28,14,22)(9,17,19,26,20,36,32,33)(10,12,24,30,25,31,23,29)(16,18,34,38,35,39,21,37);(8,12)(9,18)(10,22)(11,24)(13,30)(14,29)(15,31)(16,33)(17,34)(19,38)(20,39)(21,36)(23,28)(25,27)(26,35)(32,37)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 15682);