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×#1387 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C7 | #102 |
| commutator | C8 | C2×C42×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C24×C7 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ C23×C4×C7 ◁ C23×C7×C8 ◁ G | |
614 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2)(23,24,25,26,27,28,29);(3,4,5,6)(8,12)(9,16)(10,17)(13,15)(14,21)(18,22)(19,20);(7,8,9,13,17,19,22,20,11,15,18,12,10,14,16,21)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5556);