Finite Group DatabasebetaFinite Group Databasebeta| order | 1944 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C9 · Schur multiplier (H2) = C32 |
| direct product | Q8×C9×3−1+2 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C9 | C22×C32 |
| commutator | C2×C3 | C22×C32×C9 |
| Frattini, socle, cosocle | C2×C32 | C22×C33 |
| prime radical | C2×C33 | C22×C32 |
| LCS | G ▷ C2×C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C9 ◁ G | |
444 conjugacy classes of subgroups; only the 342 normal subgroups are shown.
This group has 444 conjugacy classes of subgroups — too many to draw. Only its 342 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 — 3 generators on 26 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)(18,19,21,20,23,25,24,22,26);(19,22,23)(21,25,26)495 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1944, 2228);