Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| isoclinism | isoclinic class of #634 |
| automorphism | Aut(G) = C24×AGL(2,𝔽3) (order 6912) · Inn(G) = C2×C3⋊D3 · Out(G) = C22×GL(2,𝔽3) |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Sylow | C32 ⋊ C4⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×C3⋊D3 |
| commutator | C2×C32 | C2×C4 |
| Fitting | C2×C32×C4 | C2 |
| socle, prime radical, cosocle | C22×C32 | C22 |
| hypercenter | C2×C4 | C3⋊D3 |
| hypocenter | C32 | C4⋊C4 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
78 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 14 points:
(1,2,3,4)(6,8)(12,13,14);(5,6,7,8)(10,11)(13,14);(9,10,11)Polycyclic presentation:
2^4,3^2|a^2=d,b^2=c,b^a=bd,e^a=e^2,f^a=f^242 conjugacy classes — view the full character table.
| maximal subgroup of | #63837, #63839, #63846, … |
| maximal normal subgroup of | #63837, #63839, #63846, … |
| maximal quotient of | #63634, #63635, #63641, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 93);