Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| isoclinism | isoclinic class of #3688 |
| automorphism | Aut(G) = C2×((C22⋊C9)⋊C2)⋊C3 (order 432) · Inn(G) = (C22⋊C9)⋊C2 · Out(G) = C2×C3 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived | Q8⋊C9 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | (C22⋊C9)⋊C2 |
| commutator, cosocle, hypocenter | Q8⋊C9 | C2 |
| Frattini, socle | C2×C3 | S4 |
| Fitting | C3×Q8 | D3 |
| LCS | G ▷ Q8⋊C9 | |
| UCS | C1 ◁ C2 | |
25 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,2,8,3,4,7,6,5)(10,12)(11,15)(13,14)(16,17);(2,5,8)(3,6,7)(9,17,12,15,14,13,11,10,16)Polycyclic presentation:
2,3^2,2^3|b^3=c^2,d^2=f,e^2=f,b^a=b^2c,c^a=c^2,d^a=e,e^a=d,d^b=ef,e^b=de,e^d=ef15 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#3689, #63689, #63691, … |
| maximal normal subgroup of | C2×#3689, #63689, #63691, … |
| maximal quotient of | #63421, C2×#3689, #88391, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 32);