Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| automorphism | Aut(G) = C2×D32×D4 (order 576) · Inn(G) = C2×D32 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C32 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D32 |
| commutator, socle, cosocle | C2×C32 | C23 |
| Fitting | C32×C4 | C22 |
| hypercenter | C4 | D32 |
| hypocenter | C32 | #41 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C4 | |
86 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 14 points:
(1,4,5,3)(2,6,7,8)(9,10,11)(12,13,14);(1,3,5,4)(2,6,7,8)(10,11);(1,2)(3,6)(4,8)(5,7)(13,14)Polycyclic presentation:
2^4,3^2|b^2=d,c^2=d,c^a=cd,f^a=f^2,e^b=e^227 conjugacy classes — view the full character table.
| maximal subgroup of | #63795, #63800, #63806, … |
| maximal normal subgroup of | #63795, #63800, #63806, … |
| maximal quotient of | #63841, #63844, #63847, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 138);