Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C8 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#13123 |
| subgroup | G/H | |
|---|---|---|
| center | C7×C8 | #121 |
| commutator | C2×C4 | C22×C7×C8 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C2×C7 | #1923 |
| prime radical | C2×C7×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C7×C8 ◁ C22×C7×C8 ◁ G | |
312 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)(8,9,13,19,14,18,34,37,15,20,35,44,12,33,22,48)(10,21,25,46,26,40,55,65,27,47,56,68,24,16,36,39)(11,28,31,59,32,58,60,69,30,17,41,43,29,42,49,62)(23,50,53,70,54,38,63,64,52,45,66,67,51,57,61,71);(8,10,22,36,12,24,35,56,15,27,34,55,14,26,13,25)(9,16,37,46,33,47,19,39,20,40,48,68,18,21,44,65)(11,23,49,61,29,51,41,66,30,52,60,63,32,54,31,53)(17,38,62,67,58,50,43,64,28,57,69,70,42,45,59,71);(8,11,12,29,15,30,14,32)(9,17,18,42,20,28,33,58)(10,23,24,51,27,52,26,54)(13,31,22,49,35,41,34,60)(16,38,21,45,40,57,47,50)(19,43,37,62,44,59,48,69)(25,53,36,61,56,66,55,63)(39,64,46,67,65,71,68,70)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6111);