Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of ((C3⋊D3)⋊C3)⋊C2 |
| automorphism | Aut(G) = C23×((C3⋊D3)⋊C3)⋊C2 (order 864) · Inn(G) = ((C3⋊D3)⋊C3)⋊C2 · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | ((C3⋊D3)⋊C3)⋊C8 |
| derived, Sylow | H(3,𝔽3) ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | ((C3⋊D3)⋊C3)⋊C2 |
| commutator, hypocenter | H(3,𝔽3) | C2×C8 |
| Frattini | C3×C4 | D32 |
| Fitting, cosocle | C4×H(3,𝔽3) | C22 |
| socle | C2×C3 | D3×Dic3 |
| prime radical | C2×(C3⋊D3)⋊C3 | C4 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C4 | |
85 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 17 points:
(2,8,6,4,5,9)(3,7)(10,11,12,14,13,15,16,17);(1,2,4)(3,5,8,7,6,9)Polycyclic presentation:
2^4,3^3|b^2=c,c^2=d,f^a=f^2,g^a=g^2,e^b=e^2,g^b=g^2,f^e=fg44 conjugacy classes — view the full character table.
| maximal subgroup of | C8×((C3⋊D3)⋊C3)⋊C2, #11720835, #11720841, … |
| maximal normal subgroup of | C8×((C3⋊D3)⋊C3)⋊C2, #11720835, #11720841, … |
| maximal quotient of | ((C3⋊D3)⋊C3)⋊C16, #11720247, #11720310, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 76);