Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13312 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#13468 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #469 |
| commutator, Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C2×C7 | #2065 |
| prime radical | C2×C7×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ C7×#405 ◁ G | |
304 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)(8,9,14,40,15,22,39,21)(10,23,55,44,29,52,28,63)(11,30,41,51,36,65,35,48)(12,20,37,18,38,53,13,19)(16,42,45,25,46,27,61,26)(17,34,49,32,50,67,54,33)(24,43,58,71,59,70,64,69)(31,66,62,47,60,68,57,56);(8,10,12,25,14,28,37,42,15,29,38,26,39,55,13,27)(9,16,18,23,21,45,20,44,22,46,19,52,40,61,53,63)(11,31,32,59,35,62,34,64,36,60,33,24,41,57,67,58)(17,47,30,43,49,66,51,69,50,56,65,70,54,68,48,71);(8,11)(9,17)(10,24)(12,34)(13,33)(14,41)(15,36)(16,43)(18,51)(19,48)(20,30)(21,54)(22,50)(23,56)(25,31)(26,60)(27,62)(28,64)(29,59)(32,37)(35,39)(38,67)(40,49)(42,57)(44,66)(45,71)(46,70)(47,52)(53,65)(55,58)(61,69)(63,68)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6456);