Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C32 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#13639 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7×C16 | C22 |
| commutator | C2 | C22×C7×C32 |
| Frattini, cosocle | C2×C16 | C23×C7 |
| socle | C22×C7 | C22×C16 |
| prime radical | C23×C7 | C2×C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7×C16 ◁ G | |
202 conjugacy classes of subgroups.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 47 points:
(1,2,3,4,5,6,7)(8,9)(10,13)(11,12)(14,15)(16,17,18,22,19,23,26,32,20,24,27,33,29,35,38,42,21,25,28,34,30,36,39,43,31,37,40,44,41,45,46,47);(9,12)(13,15);(8,10,11,14)(9,13,12,15)1120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6627);