Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7389 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C7×C8 · Schur multiplier (H2) = C2 |
| direct product | C7×#7391 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #1333 |
| commutator | C16 | C2×C7×C8 |
| Frattini, cosocle | #346 | C22×C7 |
| prime radical | C22×C7 | #361 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C4×C7×C8 ◁ C7×#346 ◁ G | |
116 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 116 conjugacy classes of subgroups — too many to draw. Only its 50 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)(8,9,12,18,14,20,36,48,15,22,13,37,35,21,39,19)(10,23,25,47,26,56,33,55,27,58,60,70,40,57,38,69)(11,28,30,52,31,53,67,51,32,65,34,17,46,49,59,50)(16,29,41,24,43,61,54,63,45,66,71,64,44,62,42,68);(8,10,15,27)(9,16,22,45)(11,29,32,66)(12,33,13,38)(14,40,35,26)(17,47,52,70)(18,42,37,54)(19,41,48,71)(20,44,21,43)(23,53,58,49)(24,59,64,67)(25,39,60,36)(28,57,65,56)(30,63,34,68)(31,62,46,61)(50,69,51,55)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 379);