Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| isoclinism | isoclinic class of #634 |
| automorphism | Aut(G) = C2×D4×AGL(2,𝔽3) (order 6912) · Inn(G) = C2×C3⋊D3 · Out(G) = C22×GL(2,𝔽3) |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| Fitting | C2×C32×C4 ⋊ C2 |
| Sylow | C32 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×C3⋊D3 |
| commutator, socle, cosocle | C2×C32 | C23 |
| Frattini | C2 | C22×C3⋊D3 |
| Fitting | C2×C32×C4 | C2 |
| hypercenter | C2×C4 | C3⋊D3 |
| hypocenter | C32 | #41 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C4 ◁ C2×C4 | |
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:
(2,6)(5,8)(12,13,14);(1,3,4,7)(2,5,6,8)(9,10,11);(1,2)(3,5)(4,6)(7,8)(10,11)(13,14)Polycyclic presentation:
2^4,3^2|b^2=d,c^a=cd,e^a=e^2,f^a=f^242 conjugacy classes — view the full character table.
| maximal subgroup of | #63570, #63632, #63652, … |
| maximal normal subgroup of | #63570, #63632, #63652, … |
| maximal quotient of | C4×#634, #64079, #64081, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 171);