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, Frattini, hypercenter | C2 | S42 |
| commutator, cosocle, hypocenter | A4×SL(2,𝔽3) | C22 |
| Fitting | C22×Q8 | D32 |
| socle | C23 | D3×S4 |
| LCS | G ▷ A4×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
327 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 327 conjugacy classes of subgroups — too many to draw. Only its 20 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^2,3^2,2^5|b^2=i,e^2=i,f^2=i,b^a=bi,c^a=c^2,g^a=gh,d^b=d^2,e^b=ef,f^b=fi,g^c=h,h^c=gh,e^d=f,f^d=efi,f^e=fi34 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157459);