Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| automorphism | Aut(G) = #11904362 (order 1152) · Inn(G) = C2×D32 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C32 ⋊ C2×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D32 |
| commutator, socle, cosocle | C2×C32 | C23 |
| Fitting | C32×C4 | C22 |
| prime radical | C2×C3⋊D3 | C22 |
| hypercenter | C4 | D32 |
| hypocenter | C32 | C2×Q8 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C4 | |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 14 points:
(1,4,5,8)(2,6,3,7)(9,10,11)(12,13,14);(1,3,5,2)(4,6,8,7)(10,11);(1,2,5,3)(4,7,8,6)(13,14)Polycyclic presentation:
2^4,3^2|a^2=d,b^2=d,c^2=d,c^a=cd,f^a=f^2,c^b=cd,e^b=e^224 conjugacy classes — view the full character table.
| maximal subgroup of | #63728, #63730, #63796, … |
| maximal normal subgroup of | #63728, #63730, #63796, … |
| maximal quotient of | #63838, #63839, #63843, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 140);