Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| isoclinism | isoclinic class of C3⋊#226 |
| automorphism | Aut(G) = C2×((C3⋊S4)⋊C3)⋊C2 (order 864) · Inn(G) = C3⋊S4 · Out(G) = C2×D3 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C3⋊#226 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C3⋊S4 |
| commutator, prime radical, cosocle, hypocenter | C3×SL(2,𝔽3) | C2 |
| Fitting | C3×Q8 | D3 |
| socle | C2×C3 | S4 |
| LCS | G ▷ C3×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,6,10,13,5,15,12,7)(2,11,14,4,8,3,16,9)(18,19);(17,18,19);(3,9,12)(4,10,11)(6,13,16)(7,14,15)Polycyclic presentation:
2,3^2,2^3|a^2=f,d^2=f,e^2=f,b^a=b^2,c^a=c^2,d^a=e,e^a=d,d^b=ef,e^b=de,e^d=ef15 conjugacy classes — view the full character table.
| maximal subgroup of | #64196, #64197, D3×#226, … |
| maximal normal subgroup of | #64196, #64197, D3×#226, … |
| maximal quotient of | C3⋊#859, C2×C3⋊#226, C9⋊#226, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 124);