Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×S4×GL(2,𝔽3) (order 2304) · Inn(G) = C22 · Out(G) = C2×D3×GL(2,𝔽3) |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C22×C3 |
| direct product | C32×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C4 | C22 |
| commutator, Frattini, cosocle | C2 | C23×C32 |
| socle | C2×C32 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32×C4 ◁ G | |
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:
(1,2)(3,6)(4,5)(7,8)(9,10,11);(2,5)(6,8)(12,13,14);(1,3,4,7)(2,6,5,8)90 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 181);