Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7540 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C2 |
| direct product | C7×#7541 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #1415 |
| commutator | C32 | C2×C4×C7 |
| Frattini, cosocle | C2×C32 | C22×C7 |
| prime radical | C2×C7×D32 | C2 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C2×C7×C8 ◁ C2×C7×C16 ◁ C2×C7×C32 ◁ G | |
94 conjugacy classes of subgroups.
| minimal degree | μ(G) = 71 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 2 generators on 71 points:
(1,2,3,4,5,6,7)(8,9,11,18,15,21,14,39)(10,22,25,49,29,16,28,40)(12,32,30,65,36,45,35,67)(13,37,31,53,38,68,34,20)(17,43,46,23,48,56,69,26)(19,51,50,24,44,59,66,33)(27,62,57,42,64,52,60,55)(41,70,47,61,54,63,71,58);(9,16)(10,23)(11,14)(12,33)(13,34)(17,44)(18,49)(19,48)(20,52)(21,22)(24,36)(25,43)(26,29)(27,63)(28,56)(30,59)(31,38)(32,41)(35,51)(37,42)(39,40)(45,54)(46,50)(47,67)(53,62)(55,68)(57,61)(58,60)(64,70)(65,71)(66,69)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 529);