Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C7 · Schur multiplier (H2) = C28 |
| direct product | C2×C7×#2983 |
| 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 | |
1978 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);(3,4,5,8,6,7,9,10)(11,12)(13,19)(14,17)(15,18)(16,21)(20,25)(22,24)(23,26);(11,13,15,21)(12,16,18,19)(14,22,23,20)(17,25,26,24);(4,7)(8,10)(11,14,15,23)(12,17,18,26)(13,20,21,22)(16,24,19,25);(12,18)(16,19)(17,26)(24,25)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53231);