Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | C4⋊#1113 |
| Fitting | C33×C4 ⋊ C22 |
| Sylow | C33 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×#1113 |
| commutator, socle, cosocle | C2×C33 | C23 |
| Fitting | C33×C4 | C22 |
| hypercenter | C4 | #1113 |
| hypocenter | C33 | C2×D4 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C4 | |
270 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 270 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 13 points:
(1,2,3,4)(5,6,7)(8,9,10)(11,12,13);(2,4)(6,7)(9,10);(6,7)(12,13)Polycyclic presentation:
2^4,3^3|c^2=d,e^a=e^2,g^a=g^2,c^b=cd,e^b=e^2,f^b=f^248 conjugacy classes — view the full character table.
| maximal subgroup of | #11722753, #11722757, #11722781, … |
| maximal normal subgroup of | #11722753, #11722757, #11722781, … |
| maximal quotient of | #11723646, #11723647, #11723648, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 691);