Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| automorphism | Aut(G) = C2×D4×AGL(2,𝔽3) (order 6912) · Inn(G) = C22×C3⋊D3 · Out(G) = C22×S4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23×C3 |
| direct product | D4×C3⋊D3 |
| Fitting | C32×D4 ⋊ C2 |
| Sylow | C32 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×C3⋊D3 |
| commutator, socle, cosocle | C2×C32 | C23 |
| Fitting | C32×D4 | C2 |
| hypercenter | D4 | C3⋊D3 |
| hypocenter | C32 | C2×D4 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ D4 | |
162 conjugacy classes of subgroups.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 3 generators on 10 points:
(6,7)(9,10);(2,4)(8,9,10);(1,2)(3,4)(5,6,7)30 conjugacy classes — view the full character table.
| maximal subgroup of | #63782, #63926, #63930, … |
| maximal normal subgroup of | #63782, #63926, #63930, … |
| maximal quotient of | #64087, C22⋊C4×C3⋊D3, #64090, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 172);