Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33557 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C25×C4 |
| direct product | C7×#33559 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C24×C7 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C2×C7×#41 ◁ G | |
1248 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3,4,5,6,7)(8,9)(10,14)(11,13)(12,15)(16,22)(17,19)(18,20)(21,23);(9,13)(10,16)(14,21)(15,20)(17,23)(19,22);(10,16)(11,18)(14,22)(19,21);(8,10,12,16)(9,14,15,22)(11,17,18,23)(13,19,20,21)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 26547);