Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #10593244 |
| commutator, cosocle, hypocenter | C3×C22⋊SL(2,𝔽3) | C22 |
| Frattini | C23 | D3×S4 |
| Fitting | C22×C3×Q8 | C2×D3 |
| socle | C23×C3 | C2×S4 |
| prime radical | C3×#5931 | C2 |
| LCS | G ▷ C3×C22⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
253 conjugacy classes of subgroups; only the 19 normal subgroups are shown.
This group has 253 conjugacy classes of subgroups — too many to draw. Only its 19 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|a^2=i,e^2=i,f^2=i,d^a=d^2,e^a=eg,f^a=fh,c^b=c^2,e^b=ef,f^b=fi,g^b=gh,e^c=f,f^c=efi,g^c=h,h^c=gh,f^e=fi36 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 154685);