Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| isoclinism | isoclinic class of #3774 |
| automorphism | Aut(G) = #10596417 (order 576) · Inn(G) = GO+(4,𝔽2) · Out(G) = D4 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C32 ⋊ D8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | GO+(4,𝔽2) |
| commutator, cosocle | C3⋊Dic3 | C22 |
| Fitting, socle | C2×C32 | D4 |
| hypocenter | C32 | D8 |
| LCS | G ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 | |
38 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 14 points:
(1,3)(2,5,6)(8,11)(9,14)(12,13);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,14)Polycyclic presentation:
2^4,3^2|c^2=d,b^a=bc,c^a=cd,e^a=e^2,c^b=cd,e^b=f,f^b=e,e^c=e^2,f^c=f^215 conjugacy classes — view the full character table.
| maximal subgroup of | #64223, #64224, #64225, … |
| maximal normal subgroup of | #64223, #64224, #64225, … |
| maximal quotient of | #63734, #63735, #63736, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 117);