Finite Group DatabasebetaFinite Group Databasebeta| order | 1944 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of C32⋊H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33×C4 · Schur multiplier (H2) = C2×C33 |
| direct product | C2×C4×#6891 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C4 | C33 |
| commutator | C32 | C2×C33×C4 |
| Frattini, cosocle | C2×C32 | C22×C33 |
| socle | C22×C32 | C2×C33 |
| prime radical | C22×C34 | C2×C3 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C4 ◁ G | |
1008 conjugacy classes of subgroups; only the 360 normal subgroups are shown.
This group has 1008 conjugacy classes of subgroups — too many to draw. Only its 360 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) = 216 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(7,8,10,9,12,14,13,11,15)(16,17,19)(18,21,23)(20,24,22);(3,4,5,6)(8,11,12)(10,14,15);(17,20,21)(19,23,24)408 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1944, 2170);