Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11855 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#11973 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#102 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C4×C8 | C23×C7 |
| prime radical | C23×C7 | #102 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ C2×C4×C7×C8 ◁ G | |
288 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 55 points:
(1,2,3,4,5,6,7)(8,9,11,15,12,14,20,22)(10,16,17,23,18,13,19,21)(24,25,29,34,27,36,43,33,30,35,46,51,44,32,28,45)(26,37,40,49,42,50,55,53,41,52,54,31,38,48,39,47);(8,10,12,18)(9,13,14,16)(11,19,20,17)(15,23,22,21)(24,26,30,41)(25,31,35,49)(27,38,44,42)(28,40,43,54)(29,39,46,55)(32,47,36,53)(33,50,45,48)(34,52,51,37);(9,14)(13,16)(15,22)(21,23)(24,27,30,44)(25,32,35,36)(26,38,41,42)(28,29,43,46)(31,47,49,53)(33,34,45,51)(37,50,52,48)(39,54,55,40)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4961);