Finite Group DatabasebetaFinite Group Databasebeta| order | 1944 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×#6856 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C1 |
| direct product | Q8×#6858 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | C22×H(3,𝔽3) |
| commutator, Frattini, prime radical, cosocle | C2×C33 | C22×C32 |
| LCS | G ▷ C2×C33 ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C33×Q8 ◁ G | |
180 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 648 |
A minimal faithful permutation representation — 2 generators on 44 points:
(1,2,3,4)(5,8,7,6)(10,12,14)(18,19,22,21,25,31,29,24,35)(20,26,30,28,23,32,39,34,42)(27,37,40,38,36,41,44,33,43);(1,5,3,7)(2,6,4,8)(9,10,13)(11,14,17)(12,16,15)(18,20,27,21,28,38,29,39,44)(19,23,33,25,34,37,24,26,36)(22,30,40,31,32,41,35,42,43)175 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1944, 647);