Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| isoclinism | isoclinic class of GO+(4,𝔽2) |
| automorphism | Aut(G) = #10596417 (order 576) · Inn(G) = GO+(4,𝔽2) · Out(G) = D4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C32 ⋊ C4⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | GO+(4,𝔽2) |
| commutator | C3⋊D3 | C2×C4 |
| Fitting, socle | C2×C32 | D4 |
| prime radical, cosocle | C2×C3⋊D3 | C22 |
| hypocenter | C32 | C4⋊C4 |
| LCS | G ▷ C3⋊D3 ▷ C32 | |
| UCS | C1 ◁ C2 | |
42 conjugacy classes of subgroups.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 10 points:
(1,3)(2,5,6)(7,9,10,8);(1,2)(3,6)(4,5)(7,8,10,9)Polycyclic presentation:
2^4,3^2|a^2=d,b^2=d,b^a=bc,e^a=e^2,e^b=f,f^b=e,e^c=e^2,f^c=f^218 conjugacy classes — view the full character table.
| maximal subgroup of | PSU(3,𝔽2)⋊C4, AGL(1,𝔽9)⋊C4, D32⋊Q8, … |
| maximal normal subgroup of | PSU(3,𝔽2)⋊C4, AGL(1,𝔽9)⋊C4, D32⋊Q8, … |
| maximal quotient of | #63732, #63733, #63737, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 116);