Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23×C3 |
No aliases for this group.
| Fitting | C23×C3×Q8 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×C3⋊#1020 |
| commutator, cosocle, hypocenter | C3×C22⋊SL(2,𝔽3) | C22 |
| Fitting | C23×C3×Q8 | D3 |
| socle | C23×C3 | C2×S4 |
| hypercenter | C22 | C3⋊#1020 |
| LCS | G ▷ C3×C22⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C22 | |
600 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 600 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^2,3^2,2^5|g^2=i,h^2=i,b^a=bi,c^a=c^2,d^a=d^2,e^a=ef,g^a=gh,h^a=hi,e^c=f,f^c=ef,g^c=h,h^c=ghi,h^g=hi48 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157783);