Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C9 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Sylow | H(7,𝔽2) ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2×C3 | C24×A4 |
| commutator, hypocenter | Q8 | C24×C9 |
| Fitting, prime radical | C3×H(7,𝔽2) | C3 |
| cosocle | C3×Q8 | C24×C3 |
| hypercenter | C3×2−1+4 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C2×C3 ◁ C3×2−1+4 | |
2324 conjugacy classes of subgroups; only the 337 normal subgroups are shown.
This group has 2324 conjugacy classes of subgroups — too many to draw. Only its 337 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:
2^4,3^2,2^3|a^2=i,b^2=i,c^2=i,e^3=f,g^2=i,h^2=i,c^a=ci,d^a=di,c^b=ci,g^e=h,h^e=ghi,h^g=hi201 conjugacy classes — view the full character table.
| maximal subgroup of | #72245141, #72245329, #72245351, … |
| maximal normal subgroup of | #72245141, #72245329, #72245351, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 153894);