Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| isoclinism | isoclinic class of AGL(1,𝔽9) |
| automorphism | Aut(G) = AGL(1,𝔽9)⋊C4 (order 288) · Inn(G) = AGL(1,𝔽9) · Out(G) = C4 |
| homology | Abelianization (H1) = C16 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | C32 ⋊ C16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | AGL(1,𝔽9) |
| commutator, hypocenter | C32 | C16 |
| Fitting, socle, prime radical | C2×C32 | C8 |
| cosocle | #622 | C2 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C2 | |
13 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 25 points:
(2,3,4,7,5,8,9,6)(10,11,12,15,13,16,18,21,14,17,19,22,20,23,24,25);(1,4,9)(2,7,3)(5,8,6)Polycyclic presentation:
2^4,3^2|a^2=b,b^2=c,c^2=d,e^a=f,f^a=ef^2,e^b=ef^2,f^b=e^2f^2,e^c=e^2,f^c=f^218 conjugacy classes — view the full character table.
| maximal subgroup of | #64213, #64214, C2×#3771, … |
| maximal normal subgroup of | #64213, #64214, C2×#3771, … |
| maximal quotient of | #63725, C2×#3771, #88586, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 114);