Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| Fitting | C2×C3×#127 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #10593631 |
| commutator, cosocle | C2×C3×C42⋊C3 | C22 |
| Frattini | C23 | C2×C3⋊S4 |
| Fitting | C2×C3×#127 | D3 |
| socle | C23×C3 | C2×S4 |
| hypocenter | C3×C42⋊C3 | D4 |
| LCS | G ▷ C2×C3×C42⋊C3 ▷ C3×C42⋊C3 | |
| UCS | C1 ◁ C2 | |
418 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 418 conjugacy classes of subgroups — too many to draw. Only its 31 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
Polycyclic presentation:
2^3,3^2,2^4|f^2=h,g^2=i,b^a=bc,d^a=d^2,e^a=e^2,f^a=fg,g^a=gi,h^a=hi,f^b=fh,g^b=gi,f^d=ghi,g^d=fgh,h^d=i,i^d=hi42 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 155051);