Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Sylow | C32 ⋊ #1412 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #10589181 |
| commutator | C32×C42 | C2×C4 |
| Frattini, hypercenter | #127 | D32 |
| Fitting, prime radical, cosocle | C32×#127 | C22 |
| socle | C2×C32 | #354 |
| hypocenter | C32 | #1412 |
| LCS | G ▷ C32×C42 ▷ C2×C32×C4 ▷ C22×C32 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #127 | |
243 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 243 conjugacy classes of subgroups — too many to draw. Only its 40 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:
2^7,3^2|a^2=d,b^2=f,c^2=fg,e^2=g,b^a=bc,c^a=ce,e^a=ef,f^a=fg,i^a=hi^2,c^b=cf,d^b=def,e^b=eg,h^b=h^2,i^b=i^2,d^c=dfg,e^d=eg54 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 6889);