Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#9031 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #120 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| socle | C22×C7 | #386 |
| prime radical | C7×#2397 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ G | |
670 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3,4,5,6,7)(9,12,14,15)(16,17,21,23,20,18,19,22);(8,9,13,12,11,14,10,15)(16,18,20,17)(19,22,21,23);(19,21)(22,23)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 2019);