Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C16 · Schur multiplier (H2) = C23×C4 |
| direct product | C7×#13189 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C7×C8 | C23 |
| commutator | C22 | C22×C7×C16 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C23×C7 | C22×C8 |
| prime radical | C22×C7×D4 | C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7×C8 ◁ G | |
534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3,4,5,6,7)(9,11)(13,15)(16,17,19,23,20,24,27,30,21,18,22,25,28,26,29,31);(8,9,10,11)(16,18,19,25,20,26,27,31,21,17,22,23,28,24,29,30);(12,13,14,15)784 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6177);