Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of #88935 |
| automorphism | Aut(G) = C22×D3×GO+(4,𝔽2) (order 1728) · Inn(G) = D32⋊D3 · Out(G) = C23 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C33 ⋊ Q16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D32⋊D3 |
| commutator, cosocle | C3×C3⋊Dic3 | C22 |
| Fitting, socle, prime radical | C2×C33 | D4 |
| hypocenter | C33 | Q16 |
| LCS | G ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 | |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,3)(4,6)(5,9,7)(10,12,14,20)(11,15,17,24)(13,22,23,19)(16,25,21,18);(2,3)(4,5)(6,7)(8,9)(10,11,14,17)(12,18,20,25)(13,21,23,16)(15,22,24,19)Polycyclic presentation:
2^4,3^3|a^2=d,b^2=d,c^2=d,b^a=bc,c^a=cd,e^a=f,f^a=e,g^a=g^2,c^b=cd,e^b=e^2,e^c=e^2,f^c=f^230 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#3776, #11724131, #11724133, … |
| maximal normal subgroup of | D3×#3776, #11724131, #11724133, … |
| maximal quotient of | #11722774, #11722775, #11722779, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 585);