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) = C34 |
| direct product | C3×Q8×C32⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C33 | C22×C32 |
| commutator | C2×C3 | C22×C32×C9 |
| Frattini, cosocle | C2×C32 | C22×C33 |
| prime radical | C2×C34 | C22×C3 |
| LCS | G ▷ C2×C3 ▷ C1 | |
| UCS | C1 ◁ C2×C33 ◁ G | |
864 conjugacy classes of subgroups; only the 432 normal subgroups are shown.
This group has 864 conjugacy classes of subgroups — too many to draw. Only its 432 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 648 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3)(22,23,24,25)(26,29,28,27);(4,5,7)(6,9,11)(8,12,10)(13,14,16,15,17,19,18,20,21)(22,26,24,28)(23,27,25,29);(5,8,9)(7,11,12)495 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1944, 2224);