Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | H(7,𝔽2) ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2×C3 | C22×C22⋊A4 |
| commutator, hypocenter | H(5,𝔽2) | C22×C9 |
| Fitting, prime radical | C3×H(7,𝔽2) | C3 |
| cosocle | C3×H(5,𝔽2) | C22×C3 |
| hypercenter | C3×D4 | C22⋊A4 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C2×C3 ◁ C3×D4 | |
2107 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 2107 conjugacy classes of subgroups — too many to draw. Only its 77 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:
2^2,3^2,2^5|a^2=i,c^3=d,b^a=bi,e^c=f,f^c=ef,g^c=h,h^c=gh,h^e=hi,g^f=gi105 conjugacy classes — view the full character table.
| maximal subgroup of | #72245141, #72245351, #72245503, … |
| maximal normal subgroup of | #72245141, #72245351, #72245503, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 153916);