Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| isoclinism | isoclinic class of D32 |
| automorphism | Aut(G) = C2×#64241 (order 576) · Inn(G) = D32 · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived, Sylow | C32 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | D32 |
| commutator, hypocenter | C32 | C2×C8 |
| Fitting, cosocle | C32×C4 | C22 |
| socle | C2×C32 | C2×C4 |
| prime radical | C2×C3⋊D3 | C4 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C4 | |
52 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 14 points:
(1,3,4,8,5,2,6,7)(10,11)(12,13,14);(1,2,4,7,5,3,6,8)(9,10,11)(13,14)Polycyclic presentation:
2^4,3^2|a^2=c,b^2=c,c^2=d,f^a=f^2,e^b=e^236 conjugacy classes — view the full character table.
| maximal subgroup of | D32⋊C8, #63729, #63730, … |
| maximal normal subgroup of | D32⋊C8, #63729, #63730, … |
| maximal quotient of | #63542, #63544, #63553, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 53);