Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33359 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C26 |
| direct product | C7×#33373 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | C22×#99 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C2×Q8 | C24×C7 |
| prime radical | C7×#3498 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C7×Q8 ◁ C2×C7×H(5,𝔽2) ◁ G | |
1312 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)(8,9,13,19,14,20,10,16)(11,17,22,15,12,18,21,23);(8,10,14,13)(9,15,20,23)(11,21,12,22)(16,18,19,17);(8,10,14,13)(9,16,20,19)(11,22,12,21)(15,18,23,17);(8,11,14,12)(9,17,20,18)(10,21,13,22)(15,16,23,19)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 26361);