Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7×C8 · Schur multiplier (H2) = C26 |
| direct product | C7×#17353 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C22×C22⋊C4 |
| commutator | C22 | C23×C7×C8 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| prime radical | C22×C7×#41 | C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C4×C7×Q8 ◁ G | |
1238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3,4,5,6,7)(8,9,10,12,11,13,14,15)(16,17,21,26,22,25,20,24)(18,27,28,31,19,23,29,30);(16,18,22,19)(17,23,25,27)(20,28,21,29)(24,30,26,31);(16,19,22,18)(17,23,25,27)(20,28,21,29)(24,31,26,30);(16,20,22,21)(17,24,25,26)(18,28,19,29)(23,30,27,31)574 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 10341);