Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| isoclinism | isoclinic class of #3713 |
| automorphism | Aut(G) = #11863964 (order 1152) · Inn(G) = #625 · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C32 ⋊ Q16 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #625 |
| commutator, Fitting, cosocle | C32×C4 | C22 |
| Frattini, hypercenter | C4 | D32 |
| socle, prime radical | C2×C32 | D4 |
| hypocenter | C32 | Q16 |
| LCS | G ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C4 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(18,19)(20,21,22);(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(17,18,19)(21,22)Polycyclic presentation:
2^4,3^2|a^2=d,b^2=d,c^2=d,b^a=bc,c^a=cd,f^a=f^2,c^b=cd,e^b=e^221 conjugacy classes — view the full character table.
| maximal subgroup of | #63735, #63736, #63802, … |
| maximal normal subgroup of | #63735, #63736, #63802, … |
| maximal quotient of | #63565, #63576, C2×#3718, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 61);