Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #64188 |
| commutator, cosocle, hypocenter | Q8⋊(Q8⋊C9) | C2 |
| Frattini, socle, prime radical | C22×C3 | #1020 |
| Fitting | C3×Q82 | D3 |
| LCS | G ▷ Q8⋊(Q8⋊C9) | |
| UCS | C1 ◁ C22 | |
163 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 163 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 1152 |
Polycyclic presentation:
2,3^2,2^6|a^2=i,b^3=c,d^2=i,e^2=i,f^2=h,g^2=h,b^a=b^2c^2,c^a=c^2,d^a=de,e^a=ei,f^a=fg,g^a=gh,d^b=e,e^b=dei,f^b=g,g^b=fgh,e^d=ei,g^f=gh42 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 154462);