Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1404 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C7×C8 · Schur multiplier (H2) = C22 |
| direct product | C7×#7347 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C2×C7×C8 |
| Frattini, cosocle | #408 | C22×C7 |
| socle | C2×C7 | #1316 |
| prime radical | C7×#585 | C4 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C2×C4×C7 ◁ C22×C4×C7 ◁ C7×#408 ◁ G | |
318 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 318 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| 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)(8,9,12,18,14,19,33,51,15,20,34,52,13,35,38,50)(10,21,23,53,24,54,58,70,25,45,59,67,36,66,31,63)(11,26,28,60,29,17,44,46,30,47,32,64,37,48,62,71)(16,39,40,22,42,55,49,57,43,65,68,27,41,61,69,56);(8,10)(9,16)(11,27)(12,31)(13,36)(14,24)(15,25)(17,45)(18,49)(19,42)(20,43)(21,48)(22,30)(23,33)(26,54)(28,39)(29,56)(32,65)(34,58)(35,41)(37,57)(38,59)(40,50)(44,55)(46,70)(47,66)(51,68)(52,69)(53,60)(61,62)(63,71)(64,67)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 335);