Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| isoclinism | isoclinic class of #3714 |
| automorphism | Aut(G) = C2×D32×D4 (order 576) · Inn(G) = #626 · Out(G) = C23 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C32 ⋊ SD16 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #626 |
| commutator, Fitting, cosocle | C32×C4 | C22 |
| Frattini, hypercenter | C4 | D32 |
| socle | C2×C32 | D4 |
| prime radical | C4⋊(C3⋊D3) | C2 |
| hypocenter | C32 | SD16 |
| LCS | G ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C4 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 14 points:
(1,3,4,2,5,6,8,7)(10,11)(12,13,14);(1,2,5,7)(3,8,6,4)(9,10,11)(13,14)Polycyclic presentation:
2^4,3^2|a^2=d,b^2=c,c^2=d,b^a=bc,c^a=cd,f^a=f^2,e^b=e^224 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#204, #63794, #63801, … |
| maximal normal subgroup of | D3×#204, #63794, #63801, … |
| maximal quotient of | #63564, #63566, #63573, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 60);