Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#12668 |
| subgroup | G/H | |
|---|---|---|
| center | C7×C8 | #106 |
| commutator | C8 | C2×C42×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C2×C7 | C4×#106 |
| prime radical | C23×C7 | #135 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C7×C8 ◁ C22×C7×C8 ◁ C2×C4×C7×C8 ◁ G | |
308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,13,19,14,21,27,34,15,20,12,18,26,33,11,17)(10,22,23,32,24,35,28,31,25,38,39,37,30,36,29,16);(8,10)(9,16,20,31)(11,23)(12,28)(13,29)(14,30)(15,25)(17,22,34,38)(18,35,19,36)(21,37,33,32)(24,26)(27,39);(8,11,26,12,15,27,14,13)(9,17,33,18,20,34,21,19)(10,23,24,28,25,39,30,29)(16,22,32,35,31,38,37,36)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5656);