Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of D32⋊D3 |
| automorphism | Aut(G) = C22×D3×GO+(4,𝔽2) (order 1728) · Inn(G) = D32⋊D3 · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C33 ⋊ C4⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D32⋊D3 |
| commutator | C3×C3⋊D3 | C2×C4 |
| Fitting, socle | C2×C33 | D4 |
| prime radical, cosocle | C2×C3×C3⋊D3 | C22 |
| hypocenter | C33 | C4⋊C4 |
| LCS | G ▷ C3×C3⋊D3 ▷ C33 | |
| UCS | C1 ◁ C2 | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 13 points:
(1,2,3)(4,6)(5,9,7)(10,12,13,11);(2,3)(4,5)(6,7)(8,9)(10,11,13,12)Polycyclic presentation:
2^4,3^3|a^2=d,b^2=d,b^a=bc,e^a=f,f^a=e,g^a=g^2,e^b=e^2,e^c=e^2,f^c=f^236 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#3773, (D32⋊D3)⋊C4, #11724115, … |
| maximal normal subgroup of | D3×#3773, (D32⋊D3)⋊C4, #11724115, … |
| maximal quotient of | #11722763, #11722764, #11722769, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 581);