Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| isoclinism | isoclinic class of C3⋊(#41⋊C3) |
| automorphism | Aut(G) = C2×D3×S4 (order 288) · Inn(G) = D3×A4 · Out(G) = C22 |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(#41⋊C3) |
| unambiguous semidirect product | #239⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×A4 |
| commutator, Fitting, cosocle, hypocenter | C3×Q8 | C2×C3 |
| socle | C2×C3 | C2×A4 |
| LCS | G ▷ C3×Q8 | |
| UCS | C1 ◁ C2 | |
33 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,5,8)(3,13,12,14,9,7,10,15,4,6,11,16)(18,19);(1,11,5,9)(2,15,8,13)(3,4,10,12)(6,7,14,16)(17,18,19)Polycyclic presentation:
2,3,2^2,3,2|a^2=f,c^2=f,d^2=f,e^a=e^2,c^b=cd,d^b=c,d^c=df21 conjugacy classes — view the full character table.
| maximal subgroup of | #64196, #64197, #64198, … |
| maximal normal subgroup of | #64196, #64197, #64198, … |
| maximal quotient of | Dic3×SL(2,𝔽3), C2×C3⋊(#41⋊C3), #88614, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 127);