Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| isoclinism | isoclinic class of #625 |
| automorphism | Aut(G) = #11856169 (order 1152) · Inn(G) = D32 · Out(G) = #120 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C32 ⋊ C4⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | D32 |
| commutator | C2×C32 | C2×C4 |
| Fitting, socle, prime radical, cosocle | C22×C32 | C22 |
| hypocenter | C32 | C4⋊C4 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C22 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 14 points:
(1,3,4,2)(6,8)(10,11)(12,13,14);(1,2,4,3)(5,6)(7,8)(9,10,11)(13,14)Polycyclic presentation:
2^4,3^2|a^2=c,b^2=c,b^a=bd,f^a=f^2,e^b=e^230 conjugacy classes — view the full character table.
| maximal subgroup of | #63739, #63740, #63742, … |
| maximal normal subgroup of | #63739, #63740, #63742, … |
| maximal quotient of | #63572, #63574, #63576, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 67);