Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17367 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C27×C4 |
| direct product | C7×#20587 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C7 |
| Frattini, cosocle | C24 | C24×C7 |
| prime radical | C22×C7×2−1+4 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C23×C7×Q8 ◁ G | |
3634 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3,4,5,6,7)(8,9)(10,11)(12,13)(14,22,25,18)(15,19)(16,20)(17,21)(23,36,39,30)(24,37,41,31)(26,32)(27,33)(28,34)(29,35)(38,43,40,42);(12,14)(13,18)(15,23)(16,24)(17,25)(19,30)(20,31)(21,22)(26,38)(27,39)(28,40)(29,41)(32,42)(33,36)(34,43)(35,37);(9,11)(12,15,17,27)(13,19,21,33)(14,23,25,39)(16,28,29,26)(18,30,22,36)(20,34,35,32)(24,40,41,38)(31,43,37,42);(12,16,17,29)(13,20,21,35)(14,24,25,41)(15,26,27,28)(18,31,22,37)(19,32,33,34)(23,38,39,40)(30,42,36,43)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 13575);