Finite Group DatabasebetaFinite Group Databasebeta| order | 1944 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C9×C27 · Schur multiplier (H2) = C9 |
| direct product | Q8×C9×C27 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C9×C27 | C22 |
| commutator | C2 | C22×C9×C27 |
| Frattini, cosocle | C2×C3×C9 | C22×C32 |
| socle, prime radical | C2×C32 | C22×C3×C9 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C9×C27 ◁ G | |
216 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 1944 |
A minimal faithful permutation representation — 2 generators on 44 points:
(1,2,3,4)(5,8,7,6)(9,10,11,12,13,14,15,16,17);(1,5,3,7)(2,6,4,8)(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44)1215 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1944, 652);