Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of #88614 |
| automorphism | Aut(G) = C2×A4×AGL(1,ℤ/9) (order 1296) · Inn(G) = D9⋊A4 · Out(G) = C2×C3 |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D9⋊A4 |
| commutator, Fitting, cosocle, hypocenter | Q8×C9 | C2×C3 |
| Frattini, socle | C2×C3 | D3×A4 |
| LCS | G ▷ Q8×C9 | |
| UCS | C1 ◁ C2 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,5,8)(3,13,12,14,9,7,10,15,4,6,11,16)(18,22,25,20,21,24)(19,23);(1,11,5,9)(2,15,8,13)(3,4,10,12)(6,7,14,16)(17,21,24,23,18,20,19,25,22)Polycyclic presentation:
2,3,2^2,3,2,3|a^2=f,c^2=f,d^2=f,e^3=g^2,e^a=e^2g,g^a=g^2,c^b=cd,d^b=c,e^b=eg,d^c=df26 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#88614, #11722408, #11722411, … |
| maximal normal subgroup of | C2×#88614, #11722408, #11722411, … |
| maximal quotient of | Dic9⋊SL(2,𝔽3), C2×#88614, #13047036, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 261);