Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11677 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#13288 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#102 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C23×C7 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ C2×C4×C7×C8 ◁ G | |
354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 47 points:
(1,2,3,4,5,6,7)(8,9)(10,14)(11,12)(13,15)(16,17)(18,28,33,43)(19,34)(20,24)(21,38)(22,27)(23,31,41,36)(25,29)(26,35)(30,40,45,47)(32,39,46,44)(37,42);(9,12)(13,14)(16,18,29,45,35,46,20,36,22,33,19,30,21,32,37,31)(17,23,24,39,38,47,25,43,27,41,42,44,26,40,34,28);(8,10,11,15)(9,13,12,14)(16,19,35,37,22,29,21,20)(17,24,38,25,27,42,26,34)(18,30,46,31,33,45,32,36)(23,39,47,43,41,44,40,28)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6276);