Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| isoclinism | isoclinic class of #3788 |
| automorphism | Aut(G) = D4×AΓL(1,𝔽9) (order 1152) · Inn(G) = C2×#157 · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Sylow | C32 ⋊ M16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×#157 |
| commutator, socle | C2×C32 | C2×C4 |
| Fitting | C32×C4 | C4 |
| prime radical | C2×C3⋊D3 | C4 |
| cosocle | C3⋊Dic3 | C22 |
| hypercenter | C4 | #157 |
| hypocenter | C32 | M16 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C4 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 14 points:
(1,3,6)(2,5,4)(7,9,11,10)(8,12,13,14);(1,2)(3,5,6,4)(7,8,10,12,11,13,9,14)Polycyclic presentation:
2^4,3^2|a^2=c,b^2=d,c^2=d,b^a=bd,e^a=ef^2,f^a=e^2f^2,e^c=e^2,f^c=f^218 conjugacy classes — view the full character table.
| maximal subgroup of | #63727, #63728, #63770, … |
| maximal normal subgroup of | #63727, #63728, #63770, … |
| maximal quotient of | C4⋊#622, #63777, #63778, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 131);