Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C16 · Schur multiplier (H2) = C23 |
| direct product | C7×#13556 |
| subgroup | G/H | |
|---|---|---|
| center | C7×C16 | C22⋊C4 |
| commutator | C22 | C22×C7×C16 |
| Frattini, cosocle | C2×C16 | C23×C7 |
| socle | C2×C7 | C2×C22⋊C16 |
| prime radical | C2×C7×#41 | C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C7×C16 ◁ C22×C7×C16 ◁ G | |
290 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)(8,9,11,17,12,18,26,39,13,19,27,40,29,41,51,63,14,16,28,35,30,36,52,58,31,37,53,59,54,60,68,70)(10,20,22,38,23,42,46,61,24,43,47,62,48,64,66,71,21,15,25,32,44,33,49,55,45,34,50,56,65,57,67,69);(8,10)(9,15)(11,25)(12,23)(13,24)(14,21)(16,20)(17,38)(18,33)(19,34)(22,28)(26,49)(27,50)(29,48)(30,44)(31,45)(32,35)(36,42)(37,43)(39,61)(40,62)(41,57)(46,52)(47,53)(51,67)(54,65)(55,58)(56,59)(60,64)(63,71)(66,68)(69,70);(10,21)(15,20)(22,25)(23,44)(24,45)(32,38)(33,42)(34,43)(46,49)(47,50)(48,65)(55,61)(56,62)(57,64)(66,67)(69,71)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6544);