Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1418 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#13625 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #357 |
| commutator | C16 | C22×C4×C7 |
| Frattini, cosocle | C2×C16 | C23×C7 |
| socle | C2×C7 | C2×#357 |
| prime radical | C2×C7×#508 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ C22×C7×C8 ◁ 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)(8,9,11,17,15,20,14,38)(10,21,24,54,26,56,22,52)(12,29,27,64,33,51,32,69)(13,34,28,50,37,47,36,19)(16,39,41,23,43,59,40,61)(18,45,44,57,48,68,66,31)(25,62,58,46,63,71,35,55)(30,49,67,70,60,53,65,42);(8,10)(9,16)(11,22)(12,30)(13,35)(14,24)(15,26)(17,40)(18,46)(19,49)(20,43)(21,51)(23,57)(25,36)(27,65)(28,63)(29,56)(31,61)(32,67)(33,60)(34,42)(37,58)(38,41)(39,68)(44,62)(45,59)(47,70)(48,55)(50,53)(52,69)(54,64)(66,71);(8,11,15,14)(9,17,20,38)(10,22,26,24)(12,27,33,32)(13,28,37,36)(16,40,43,41)(18,44,48,66)(19,34,50,47)(21,52,56,54)(23,39,61,59)(25,35,63,58)(29,64,51,69)(30,65,60,67)(31,45,57,68)(42,53,70,49)(46,62,55,71)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6613);