Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
No aliases for this group.
| derived | C3×#997 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×#1020 |
| commutator, cosocle, hypocenter | C3×#997 | C22 |
| Fitting | C3×H(5,𝔽2) | C2×D3 |
| socle | C2×C3 | C2×#1020 |
| LCS | G ▷ C3×#997 | |
| UCS | C1 ◁ C2 | |
564 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 564 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) = 48 |
Polycyclic presentation:
2^2,3^2,2^5|a^2=i,g^2=i,h^2=i,b^a=bi,d^a=d^2,c^b=c^2,e^b=ef,g^b=gh,h^b=hi,e^c=f,f^c=ef,g^c=h,h^c=ghi,h^e=hi,g^f=gi,h^g=hi36 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#87522, #72215533, #72215563, … |
| maximal normal subgroup of | D3×#87522, #72215533, #72215563, … |
| maximal quotient of | C3×#11923646, #72226748 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157657);