Finite Group DatabasebetaFinite Group Databasebeta| order | 1944 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of C32⋊H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C33 · Schur multiplier (H2) = C23×C33 |
| direct product | C23×C9⋊H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C32 | C33 |
| commutator, Frattini, cosocle | C32 | C23×C33 |
| prime radical | C23×C3×H(3,𝔽3) | C3 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C23×C32 ◁ G | |
1328 conjugacy classes of subgroups; only the 624 normal subgroups are shown.
This group has 1328 conjugacy classes of subgroups — too many to draw. Only its 624 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 648 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(8,10,13)(11,15,14)(16,17,19)(18,21,23)(20,24,22);(3,4)(7,8,11,9,10,14,12,13,15);(5,6)(17,20,21)(19,23,24)408 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1944, 2954);