Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C33 |
No aliases for this group.
| Fitting | C33×Q8 ⋊ C2 |
| Sylow | C33 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×C3⋊(C3⋊D3) |
| commutator, socle, cosocle | C2×C33 | C23 |
| Fitting | C33×Q8 | C2 |
| hypercenter | Q8 | C3⋊(C3⋊D3) |
| hypocenter | C33 | #41 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
560 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 216 |
A minimal faithful permutation representation — 4 generators on 17 points:
(1,3,4,7)(2,5,6,8)(15,16,17);(1,2,4,6)(3,8,7,5)(12,13,14);(3,7)(5,8)(10,11)(13,14)(16,17);(9,10,11)Polycyclic presentation:
2^4,3^3|b^2=d,c^2=d,c^a=cd,e^a=e^2,f^a=f^2,g^a=g^2,c^b=cd75 conjugacy classes — view the full character table.
| maximal subgroup of | #11723572, #11723578, #11723587, … |
| maximal normal subgroup of | #11723572, #11723578, #11723587, … |
| maximal quotient of | #11723931, #11723933, #11723934, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 727);