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) = C3 |
| direct product | Q8×#6857 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | C22×H(3,𝔽3) |
| commutator, Frattini, cosocle | C2×C33 | C22×C32 |
| prime radical | C2×C3×H(3,𝔽3) | C22×C3 |
| LCS | G ▷ C2×C33 ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C33×Q8 ◁ G | |
276 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 276 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 648 |
A minimal faithful permutation representation — 2 generators on 26 points:
(1,2,3,4)(5,8,7,6)(9,10,12,11,13,16,15,14,17)(19,21,23);(1,5,3,7)(2,6,4,8)(10,13,14)(18,19,22)(20,23,26)(21,25,24)175 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1944, 646);