Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| isoclinism | isoclinic class of #3753 |
| automorphism | Aut(G) = C2×D4×AGL(2,𝔽3) (order 6912) · Inn(G) = #638 · Out(G) = C2×GL(2,𝔽3) |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Sylow | C32 ⋊ Q16 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #638 |
| commutator, cosocle | C32×C4 | C22 |
| Frattini | C4 | C2×C3⋊D3 |
| Fitting | C32×Q8 | C2 |
| socle, prime radical | C2×C32 | D4 |
| hypercenter | Q8 | C3⋊D3 |
| hypocenter | C32 | Q16 |
| LCS | G ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q8 | |
54 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(20,21,22);(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(18,19)(21,22);(17,18,19)Polycyclic presentation:
2^4,3^2|a^2=d,b^2=d,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^2,c^b=cd27 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#216, #63943, #63946, … |
| maximal normal subgroup of | D3×#216, #63943, #63946, … |
| maximal quotient of | #63634, #63637, #63662, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 99);