Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| automorphism | Aut(G) = C2×S4×AGL(2,𝔽3) (order 20736) · Inn(G) = C22×C3⋊D3 · Out(G) = C2×D3×S4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| Fitting | C32×Q8 ⋊ C2 |
| Sylow | C32 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×C3⋊D3 |
| commutator, socle, cosocle | C2×C32 | C23 |
| Fitting | C32×Q8 | C2 |
| hypercenter | Q8 | C3⋊D3 |
| hypocenter | C32 | #41 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 14 points:
(1,3,4,7)(2,5,6,8)(12,13,14);(1,2,4,6)(3,8,7,5)(9,10,11);(3,7)(5,8)(10,11)(13,14)Polycyclic presentation:
2^4,3^2|b^2=d,c^2=d,c^a=cd,e^a=e^2,f^a=f^2,c^b=cd30 conjugacy classes — view the full character table.
| derived subgroup of | #10596520⋊C3, #10596521⋊C3, #10596522⋊C3 |
| maximal subgroup of | #63785, #63941, #63945, … |
| maximal normal subgroup of | #63785, #63941, #63945, … |
| maximal quotient of | #64099, #64101, #64102, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 175);