Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C7 · Schur multiplier (H2) = C28 |
| direct product | C2×C7×#2962 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C7 | C24 |
| commutator | C22 | C24×C4×C7 |
| Frattini, cosocle | C2×C4 | C25×C7 |
| socle | C23×C7 | C23×C4 |
| prime radical | C24×C7 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C7 ◁ G | |
1938 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 5 generators on 33 points:
(1,2)(27,28,29,30,31,32,33);(3,4)(5,10)(6,8)(7,9)(11,12,14,17,15,18,21,24)(13,16,20,23,19,22,25,26);(3,5,7,6)(4,8,9,10);(3,6,7,5)(4,8,9,10)(11,13,14,20,15,19,21,25)(12,16,17,23,18,22,24,26);(13,19)(16,22)(20,25)(23,26)700 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53177);