Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| isoclinism | isoclinic class of #3774 |
| automorphism | Aut(G) = #10596417 (order 576) · Inn(G) = GO+(4,𝔽2) · Out(G) = D4 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C32 ⋊ Q16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | GO+(4,𝔽2) |
| commutator, cosocle | C3⋊Dic3 | C22 |
| Fitting, socle, prime radical | C2×C32 | D4 |
| hypocenter | C32 | Q16 |
| LCS | G ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 | |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,3)(2,5,6)(7,9,11,17)(8,12,14,21)(10,19,20,16)(13,22,18,15);(1,2)(3,6)(4,5)(7,8,11,14)(9,15,17,22)(10,18,20,13)(12,19,21,16)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,c^b=cd,e^b=f,f^b=e,e^c=e^2,f^c=f^215 conjugacy classes — view the full character table.
| maximal subgroup of | #64223, #64226, #64228, … |
| maximal normal subgroup of | #64223, #64226, #64228, … |
| maximal quotient of | #63740, #63743, C2×#3776, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 119);