Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C8 · Schur multiplier (H2) = C23 |
| direct product | C2×C7×#1372 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C7 | D8 |
| commutator | C8 | C22×C7×C8 |
| Frattini, cosocle | C4×C8 | C23×C7 |
| socle, prime radical | C23×C7 | #107 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C7 ◁ C2×C42×C7 ◁ C2×C4×C7×C8 ◁ G | |
262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 1792 |
A minimal faithful permutation representation — 3 generators on 33 points:
(1,2)(27,28,29,30,31,32,33);(3,4,5,7,6,8,9,10)(12,16)(13,20)(14,21)(17,22)(18,26)(19,25)(23,24);(11,12,17,26,14,18,13,16,15,19,20,24,21,23,22,25)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4995);