Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| isoclinism | isoclinic class of PSU(3,𝔽2) |
| automorphism | Aut(G) = #10596418 (order 576) · Inn(G) = PSU(3,𝔽2) · Out(G) = D4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Sylow | C32 ⋊ C4⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | PSU(3,𝔽2) |
| commutator | C3⋊D3 | C2×C4 |
| Fitting, socle | C2×C32 | Q8 |
| prime radical, cosocle | C2×C3⋊D3 | C22 |
| hypocenter | C32 | C4⋊C4 |
| LCS | G ▷ C3⋊D3 ▷ C32 | |
| UCS | C1 ◁ C2 | |
31 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 13 points:
(1,4,5,6)(3,8,7,9)(10,11,12,13);(2,4,5,9)(3,7,8,6)Polycyclic presentation:
2^4,3^2|a^2=c,b^2=cd,b^a=bc,e^a=ef^2,f^a=e^2f^2,e^b=f,f^b=e^2,e^c=e^2,f^c=f^2| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | C11 | C12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 1 | 8 | 8 | 9 | 9 | 18 | 18 | 18 | 18 | 18 | 18 |
| order | 1 | 2 | 3 | 6 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 |
| χ3 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 |
| χ4 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 |
| χ5 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | ||||
| χ6 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | ||||
| χ7 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | ||||
| χ8 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | ||||
| χ9 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 |
| χ10 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| χ11 | 8 | 8 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| χ12 | 8 | -8 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Here i is the imaginary unit (i² = −1).
| maximal subgroup of | GO+(4,𝔽2)⋊C4, AGL(1,𝔽9)⋊C4, #64243, … |
| maximal normal subgroup of | GO+(4,𝔽2)⋊C4, AGL(1,𝔽9)⋊C4, #64243, … |
| maximal quotient of | #63744, #63745, #63746, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 120);