Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12026 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C2 |
| direct product | C7×#12026 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #427 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C4×C8 | C23×C7 |
| socle | C2×C7 | #1568 |
| prime radical | C23×C7 | #108 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C7×#97 ◁ C7×#405 ◁ G | |
180 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 106 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 — 3 generators on 71 points:
(1,2,3,4,5,6,7)(8,9,13,20,15,21,38,50,16,22,37,19,14,41,12,36)(10,23,26,43,28,52,25,44,29,48,39,65,27,61,59,17)(11,30,32,63,34,47,31,49,35,64,40,66,33,18,45,46)(24,53,55,42,57,67,69,62,58,71,60,68,56,51,54,70);(8,10,12,25,14,27,37,26,16,29,38,59,15,28,13,39)(9,17,19,23,21,43,36,52,22,44,20,48,41,65,50,61)(11,24,31,54,33,56,32,60,35,58,45,69,34,57,40,55)(18,42,46,51,30,62,63,53,47,68,49,67,64,70,66,71);(8,11,14,33,16,35,15,34)(9,18,21,30,22,47,41,64)(10,24,27,56,29,58,28,57)(12,31,37,32,38,45,13,40)(17,42,43,62,44,68,65,70)(19,46,36,63,20,49,50,66)(23,51,52,53,48,67,61,71)(25,54,26,60,59,69,39,55)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5014);