Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #2263 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#13636 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | C2×D16 |
| commutator | C16 | C22×C4×C7 |
| Frattini, cosocle | C2×C16 | C23×C7 |
| socle | C2×C7 | C2×#366 |
| prime radical | C23×C7 | Q32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ C22×C7×C8 ◁ C22×C7×C16 ◁ G | |
178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 71 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 71 points:
(1,2,3,4,5,6,7)(8,9,12,18,15,20,14,36)(10,21,25,47,23,45,27,58)(11,28,30,59,32,61,31,62)(13,34,33,44,35,63,29,19)(16,37,39,52,38,55,64,24)(17,41,42,51,43,50,60,22)(26,57,56,68,53,71,54,48)(40,49,46,65,66,69,67,70);(8,10,22,49,13,26,30,37,12,25,41,65,33,56,32,52,15,23,51,69,35,53,31,55,14,27,50,70,29,54,11,24)(9,16,28,48,19,40,60,58,36,64,62,71,63,67,43,45,20,38,61,68,44,66,42,47,18,39,59,57,34,46,17,21);(10,23)(16,38)(21,45)(24,52)(25,27)(26,53)(37,55)(39,64)(40,66)(46,67)(47,58)(48,68)(49,69)(54,56)(57,71)(65,70)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6624);