Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#22695 |
| 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 | |
808 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)(8,9,13,20,15,21,14,19)(10,22,26,34,27,38,25,16)(11,28,29,36,30,33,24,17)(12,18,32,37,23,35,31,39);(8,10)(9,16)(11,12)(13,26)(14,25)(15,27)(17,35)(18,36)(19,38)(20,22)(21,34)(23,30)(24,31)(28,39)(29,32)(33,37);(8,11,14,24,15,30,13,29)(9,17,20,28,21,36,19,33)(10,23,25,32,27,12,26,31)(16,18,22,37,34,35,38,39);(8,12)(9,18)(10,24)(11,26)(13,32)(14,31)(15,23)(16,33)(17,22)(19,39)(20,37)(21,35)(25,30)(27,29)(28,34)(36,38)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 15683);