Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #85554⋊C3 |
| Fitting | Q82 ⋊ C3×D3 |
| derived | Q8⋊SL(2,𝔽3) ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #5934⋊C3 |
| commutator, cosocle, hypocenter | Q8⋊SL(2,𝔽3) | C2×C3 |
| Frattini, hypercenter | C22 | #1020⋊C3 |
| Fitting | Q82 | C3×D3 |
| LCS | G ▷ Q8⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C22 | |
103 conjugacy classes of subgroups; only the 9 normal subgroups are shown.
This group has 103 conjugacy classes of subgroups — too many to draw. Only its 9 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
Polycyclic presentation:
2,3^2,2^6|d^2=h,e^2=h,f^2=hi,g^2=hi,c^a=c^2,d^a=fhi,e^a=ghi,f^a=dh,g^a=eh,h^a=hi,d^b=deh,e^b=d,f^b=fg,g^b=f,d^c=e,e^c=deh,f^c=fg,g^c=f,e^d=eh,g^f=ghi35 conjugacy classes — view the full character table.
| maximal subgroup of | Sp(4,𝔽3) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 155468);