Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C24 |
| direct product | C2×C7×#1388 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #102 |
| commutator | C8 | C2×C42×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C22×C7 | #340 |
| prime radical | C24×C7 | Q16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C23×C4×C7 ◁ C23×C7×C8 ◁ G | |
554 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2)(35,36,37,38,39,40,41);(3,4,7,13,10,15,9,26)(5,16,18,28,21,30,20,34)(6,22,24,31,25,14,8,12)(11,19,27,23,29,33,32,17);(3,5,8,19,9,20,25,17,10,21,24,33,7,18,6,23)(4,11,14,28,13,27,12,30,15,29,22,34,26,32,31,16)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5558);