Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C32 ⋊ Q128 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #10590024 |
| commutator, cosocle | C3⋊(C3⋊C32) | C22 |
| Frattini, hypercenter | C16 | GO+(4,𝔽2) |
| Fitting | C32×C16 | D4 |
| socle, prime radical | C2×C32 | D32 |
| hypocenter | C32 | Q128 |
| LCS | G ▷ C3⋊(C3⋊C32) ▷ C32×C16 ▷ C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 | |
66 conjugacy classes of subgroups.
| transitive degree | μt(G) = 384 |
Polycyclic presentation:
2^7,3^2|a^2=g,b^2=g,c^2=de,d^2=ef,e^2=fg,f^2=g,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,i^a=hi^2,c^b=cd,d^b=de,e^b=ef,f^b=fg,h^b=hi,i^b=i^2,h^c=h^2,i^c=i^257 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 7147);