Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| isoclinism | isoclinic class of (#857⋊C2)⋊D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | (#857⋊C2)⋊D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle | C3×#127⋊C3 | C22 |
| Frattini | C23 | D3×S4 |
| Fitting | C3×#127 | C2×D3 |
| socle | C22×C3 | GL(2,ℤ/4) |
| hypocenter | C3×C42⋊C3 | D4 |
| LCS | G ▷ C3×#127⋊C3 ▷ C3×C42⋊C3 | |
| UCS | C1 | |
243 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 243 conjugacy classes of subgroups — too many to draw. Only its 21 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^3,3^2,2^4|f^2=h,g^2=i,b^a=bc,e^a=e^2,f^a=fi,g^a=ghi,d^b=d^2,f^b=fgi,g^b=gi,h^b=hi,f^c=fh,g^c=gi,f^d=gh,g^d=fg,h^d=i,i^d=hi30 conjugacy classes — view the full character table.
| maximal subgroup of | #72219818, D3×#4626⋊C2 |
| maximal normal subgroup of | D3×#4626⋊C2 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 154584);