Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of C9⋊SL(2,𝔽3) |
| automorphism | Aut(G) = C2×A4⋊#272 (order 1296) · Inn(G) = C3×A4 · Out(G) = C2×C3⋊D3 |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | #41 ⋊ 3−1+2 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C3×A4 |
| commutator, cosocle | C3×Q8 | C2×C32 |
| Frattini, socle | C2×C3 | C2×C3×A4 |
| Fitting, prime radical | C32×#41 | C3 |
| hypercenter | C32×C4 | A4 |
| hypocenter | Q8 | C2×3−1+2 |
| LCS | G ▷ C3×Q8 ▷ Q8 | |
| UCS | C1 ◁ C3×C4 ◁ C32×C4 | |
70 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,15)(2,9)(3,7)(4,14)(5,13)(6,12)(8,11)(10,16)(18,21,22)(20,24,25);(3,4,9)(6,7,13)(10,12,11)(14,16,15)(17,22,25,19,21,20,23,18,24)Polycyclic presentation:
2,3^3,2^3|a^2=g,b^3=d,e^2=g,f^2=g,c^b=cd,e^b=f,f^b=ef,f^e=fg62 conjugacy classes — view the full character table.
| maximal subgroup of | #63422⋊C3, #11721359, #63690⋊C3, … |
| maximal normal subgroup of | #63422⋊C3, #11721359, #63690⋊C3, … |
| maximal quotient of | C4×#6204, C2×#88691, (#41⋊C9)⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 338);