Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×(C22⋊C9)⋊C2 |
| commutator, cosocle | C2×Q8⋊C9 | C23 |
| Frattini, socle | C22×C3 | C22×S4 |
| Fitting | C3×C4×#41 | D3 |
| hypercenter | C42 | (C22⋊C9)⋊C2 |
| hypocenter | Q8⋊C9 | C2×D4 |
| LCS | G ▷ C2×Q8⋊C9 ▷ Q8⋊C9 | |
| UCS | C1 ◁ C22 ◁ C42 | |
577 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 577 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^4,3^2,2^3|b^2=i,c^2=d,e^3=f,g^2=i,h^2=i,b^a=bi,c^a=cd,e^a=e^2f^2,f^a=f^2,g^a=gh,h^a=hi,g^e=h,h^e=ghi,h^g=hi96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 154256);