Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2196 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C26 |
| direct product | C7×#33546 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C24×C7 |
| socle | C2×C7 | C2×B(4,𝔽2) |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C2×C4×C7 ◁ C2×C7×#41 ◁ G | |
1236 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)(16,19)(17,21)(18,20)(22,23);(9,13)(14,19)(15,20)(21,23);(11,18)(16,22);(8,10,12,17)(9,14,15,21)(11,16,18,22)(13,19,20,23)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 26534);