Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C16 · Schur multiplier (H2) = C23 |
| direct product | C7×#13558 |
| subgroup | G/H | |
|---|---|---|
| center | C7×C16 | C2×D4 |
| commutator | C4 | C22×C7×C16 |
| Frattini, cosocle | C2×C16 | C23×C7 |
| socle | C2×C7 | C2×C22⋊C16 |
| prime radical | C2×C7×#41 | C8 |
| LCS | G ▷ C4 ▷ 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)(8,9,12,19,13,20,31,46,14,21,32,47,34,48,59,68,15,17,33,40,35,41,60,64,11,18,28,43,29,44,56,66)(10,22,24,49,25,50,53,69,23,37,51,45,52,62,70,67,26,42,30,57,54,65,58,71,27,16,36,38,55,39,61,63);(8,10)(9,16)(11,27)(12,30)(13,25)(14,23)(15,26)(17,37)(18,42)(19,45)(20,39)(21,22)(24,33)(28,51)(29,55)(31,58)(32,36)(34,52)(35,54)(38,40)(41,62)(43,49)(44,65)(46,67)(47,57)(48,50)(53,60)(56,70)(59,61)(63,64)(66,69)(68,71);(9,17)(16,37)(18,21)(19,40)(20,41)(22,42)(38,45)(39,62)(43,47)(44,48)(46,64)(49,57)(50,65)(63,67)(66,68)(69,71)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6546);