Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #10593244 |
| commutator, cosocle | C2×C3×C22⋊A4 | C22 |
| Frattini | C23 | D3×S4 |
| Fitting | C25×C3 | C2×D3 |
| socle | C23×C3 | C2×S4 |
| hypocenter | C3×C22⋊A4 | D4 |
| LCS | G ▷ C2×C3×C22⋊A4 ▷ C3×C22⋊A4 | |
| UCS | C1 ◁ C2 | |
521 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 521 conjugacy classes of subgroups — too many to draw. Only its 23 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|b^2=c,b^a=bc,e^a=e^2,f^a=fh,g^a=gi,d^b=d^2,f^b=fg,h^b=hi,f^d=g,g^d=fg,h^d=i,i^d=hi42 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#73081, #72229182, #72229194, … |
| maximal normal subgroup of | D3×#73081, #72229182, #72229194, … |
| maximal quotient of | C2×#11920752 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 154763);